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"P\370\355klad 1" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 " Naleznete obecne reseni rovnice " }{TEXT 258 5 "y\" + " }{TEXT 268 1 " 2" }{TEXT 267 5 "y' - " }{TEXT 270 1 "3" }{TEXT 269 4 "y = " }{TEXT -1 2 "0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "rovnice1:=diff(y(x),x$2)+2*d iff(y(x),x)-3*y(x)=0;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%)rovnice1G/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&F1F2-F( 6$F*F-F2F2*&\"\"$F2F*F2!\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Jedna se o homogenni rovnici druheho radu. Jeji reseni hledame ve \+ tvaru " }{TEXT 280 1 "y" }{TEXT -1 1 "(" }{TEXT 279 1 "x" }{TEXT -1 2 ")=" }{XPPEDIT 18 0 "exp(lambda*x);" "6#-%$expG6#*&%'lambdaG\"\"\"%\"x GF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "tvar_reseni:=y(x)=ex p(lambda*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,tvar_reseniG/-%\"yG 6#%\"xG-%$expG6#*&%'lambdaG\"\"\"F)F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 " Dosazenim do zadane rovnice dostaneme po upravach charak teristickou rovnici." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sub s(tvar_reseni,rovnice1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%%diff G6$-%$expG6#*&%'lambdaG\"\"\"%\"xGF--%\"$G6$F.\"\"#F-*&F2F--F&6$F(F.F- F-*&\"\"$F-F(F-!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&)%'lambdaG\" \"#\"\"\"-%$expG6#*&F'F)%\"xGF)F)F)*(F(F)F'F)F*F)F)*&\"\"$F)F*F)!\"\" \"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Protoze vyraz " } {XPPEDIT 18 0 "exp(lambda*x);" "6#-%$expG6#*&%'lambdaG\"\"\"%\"xGF(" } {TEXT -1 52 " je vzdy vetsi nez nula, muzeme jim rovnici vydelit." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "char_rovnice:=simplify(%/exp (lambda*x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-char_rovniceG/,(*$) %'lambdaG\"\"#\"\"\"F+*&F*F+F)F+F+\"\"$!\"\"\"\"!" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 43 " Vypocteme koreny charakteristicke rovnice." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solve(char_rovnice,lambda); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Charakteristicka rovnice ma tedy 2 realne ruzne kore ny." }}{PARA 0 "" 0 "" {TEXT -1 36 "Fundamentalni system reseni ma tva r:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "reseni1:=subs(lambda= 1,tvar_reseni); reseni2:=subs(lambda=-3,tvar_reseni);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(reseni1G/-%\"yG6#%\"xG-%$expGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(reseni2G/-%\"yG6#%\"xG-%$expG6#,$F)!\"$" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Urcime " }{TEXT 274 10 "wronskian \+ " }{TEXT -1 76 "nalezenych reseni pomoci stejnojmenneho prikazu, ktery je soucasti knihovny " }{TEXT 275 6 "linalg" }{TEXT -1 1 "," }{TEXT 276 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "with(linalg): w ronskian([rhs(reseni1),rhs(reseni2)],x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$-%$expG6#%\"xG-F)6#,$F+!\"$7$F(,$F,F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "a vypocteme ho." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$expG6#%\"xG\"\"\"-F&6#,$F(!\"$F)!\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Protoze wronskian je pro vsechna realna \+ " }{TEXT 277 1 "x" }{TEXT -1 46 " nenulovy, jsou obe reseni linearne n ezavisla." }}{PARA 0 "" 0 "" {TEXT -1 62 "Linearni kombinaci nalezenyc h reseni dostaneme obecne reseni. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "ob_reseni:=y(x)=c[1]*rhs(reseni1)+c[2]*rhs(reseni2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*ob_reseniG/-%\"yG6#%\"xG,&*&&% \"cG6#\"\"\"F/-%$expGF(F/F/*&&F-6#\"\"#F/-F16#,$F)!\"$F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Vysledek porovname s vystupem prikazu " } {TEXT 278 6 "dsolve" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "dsolve(rovnice1,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&%$_C1G\"\"\"-%$expGF&F+F+*&%$_C2GF+-F-6#,$F'! \"$F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Spravnost vypoctu over ime dosazenim obecneho reseni do puvodni rovnice." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 25 "subs(ob_reseni,rovnice1);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,*-%%diffG6$,&*&&%\"cG6#\"\"\"F--%$expG6#%\"xGF-F-*& &F+6#\"\"#F--F/6#,$F1!\"$F-F--%\"$G6$F1F5F-*&F5F--F&6$F(F1F-F-*(\"\"$F -F*F-F.F-!\"\"*(FAF-F3F-F6F-FB\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\" !F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Do jednoho obrazku zobrazi me nekolik partikularnich reseni pro ruzne hodnoty konstant " } {XPPEDIT 18 0 "c[1];" "6#&%\"cG6#\"\"\"" }{TEXT -1 1 "," }{XPPEDIT 18 0 "c[2];" "6#&%\"cG6#\"\"#" }{TEXT -1 2 ". 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\"\"F_[lFi[o" 1 2 0 1 10 1 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "Dale zobrazime grafy partikularni ch reseni prochazejicich bodem [0,0]. Do obecneho reseni tedy dosadime odpovidajici pocatecni podminku " }{TEXT 259 1 "y" }{TEXT 272 4 "(0) \+ " }{TEXT 271 2 "= " }{TEXT 273 1 "0" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "dosazeni1:=0=subs(x=0,rhs(ob_reseni));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*dosazeni1G/\"\"!,&*&&%\"cG6#\"\"\" F,-%$expG6#F&F,F,*&&F*6#\"\"#F,F-F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "a vyjadrime jednu konstantu pomoci druhe." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "konst1:=solve(dosazeni1,\{c[1]\}); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'konst1G<#/&%\"cG6#\"\"\",$&F(6# \"\"#!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Do obecneho reseni \+ dosadime vyjadreni konstanty " }{XPPEDIT 18 0 "c[1];" "6#&%\"cG6#\"\" \"" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "part VBode:=y(x)=subs(konst1,rhs(ob_reseni));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*partVBodeG/-%\"yG6#%\"xG,&*&&%\"cG6#\"\"#\"\"\"-%$expGF(F0!\" \"*&F,F0-F26#,$F)!\"$F0F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Urci me rovnici tecny ke krivce tohoto reseni v bode " }{XPPEDIT 259 0 "x[0 ];" "6#&%\"xG6#\"\"!" }{TEXT -1 14 " = 0 ve tvaru " }{TEXT 261 10 "y = kx + q" }{TEXT -1 11 ". Smernici " }{TEXT 262 1 "k" }{TEXT -1 33 " zj istime pomoci derivace v bode " }{XPPEDIT 261 0 "x[0];" "6#&%\"xG6#\" \"!" }{TEXT -1 7 ", usek " }{TEXT 266 1 "q" }{TEXT -1 26 " je funkcni \+ hodnota v bode" }{TEXT 264 1 " " }{XPPEDIT 263 0 "x[0];" "6#&%\"xG6#\" \"!" }{TEXT -1 3 ". " }{TEXT 260 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "tecna1:=y(x)=simplify(subs(x=0,diff(rhs(partVBode),x) )*x+subs(x=0,rhs(partVBode)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'t ecna1G/-%\"yG6#%\"xG,$*&F)\"\"\"&%\"cG6#\"\"#F,!\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Nyni budeme animovat partikularni reseni procha zejici bodem [0,0] spolecne s jeho tecnou v " }{XPPEDIT 265 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 27 " = 0 pro hodnotu konstanty " } {XPPEDIT 18 0 "c[2];" "6#&%\"cG6#\"\"#" }{TEXT -1 93 " menici se od -2 do 2. Vsimneme si, ze smernice tecny je pro ruzna partikularni reseni ruzna." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 208 "krivka1:=animat e(rhs(partVBode),x=-1..3,c[2]=-2..2,color=blue,frames=80): krivka2:=an imate(rhs(tecna1),x=-1..1.5,c[2]=-2..2,color=red,frames=80): \+ display(\{krivka1,krivka2\},thickness=2);" }}{PARA 13 " " 1 "" {GLPLOT2D 354 266 266 {PLOTDATA 2 "6&-%(ANIMATEG6\\p7$-%'CURVES G6$7S7$$!\"\"\"\"!$!\")F.7$$!+nm;z%*!#5$!3/LLLLLL$e(!#<7$$!+LLLe*)F4$! 31mmmmmmmrF77$$!+++]P%)F4$!37************\\nF77$$!+nmm;zF4$!3:KLLLLLLj F77$$!+LL$eR(F4$!3>lmmmmm;fF77$$!++++voF4$!3A)************\\&F77$$!+nm ;ajF4$!3EJLLLLL$3&F77$$!+LLLLeF4$!3IkmmmmmmYF77$$!+++]7`F4$!3M(******* ****\\UF77$$!+nmm\"z%F4$!3\"3LLLLLL$QF77$$!+LL$3F%F4$!3Ikmmmmm;MF77$$! ++++]PF4$!3y(*************HF77$$!+nm;HKF4$!3EJLLLLL$e#F77$$!+LLL3FF4$! 3ukmmmmmm@F77$$!+++](=#F4$!3+)***********\\7$$\")nmmT!\"*$\"3NbLLLLLLLFbq7$$\")++v$*Fa r$\"3?A++++++vFbq7$$\"*LL$e9Far$\"3'*ommmmmm6F77$$\"*nm\"z>Far$\"3qNLL 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agjn7$FhalFagjn7$F[blFagjn7$F^blFagjn7$FablFagjn7$FdblFagjn7$FgsFagjn7 $FhblFagjn7$F[clFagjn7$F^clFagjn7$FaclFagjn7$FdclFagjn7$FgclFagjn7$Fjc lFagjn7$F`uFagjnF]dl-%*THICKNESSG6#\"\"#-%+AXESLABELSG6%%\"xG%!G%(DEFA ULTG-%%VIEWG6$;F,$\"\"$F.F\\b[o" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "P\370\355klad 2" }} {EXCHG {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 " Urcete partikularni reseni rovnice " }{TEXT 282 5 "y\" + " }{TEXT 293 1 "4" }{TEXT 292 5 "y' + " }{TEXT 295 1 "5" }{TEXT 294 4 "y = " } {TEXT 302 1 "0" }{TEXT -1 24 " pro pocatecni podminky " }{TEXT 285 1 " y" }{TEXT 298 3 "(0)" }{TEXT 296 3 " = " }{TEXT 300 2 "-1" }{TEXT 299 4 ", y'" }{TEXT 297 4 "(0) " }{TEXT 286 2 "= " }{TEXT 301 1 "5" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "rovnice2:=diff(y(x),x$2)+4 *diff(y(x),x)+5*y(x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)rovnice2 G/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&\"\"%F2-F(6$F*F-F2F2 *&\"\"&F2F*F2F2\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Jde o hom ogenni rovnici druheho radu. Charakteristickou rovnici urcime pouzitim procedury " }{TEXT 290 4 "char" }{TEXT -1 12 " ze souboru " }{TEXT 291 9 "difproc.m" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "read \" difproc.m\": char_rovnice:=char(rovnice2,y(x));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%-char_rovniceG/,(*$)%'lambdaG\"\"#\"\"\"F+*&\" \"%F+F)F+F+\"\"&F+\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "a nale zneme jeji koreny. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solv e(char_rovnice,lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$^$!\"#\"\" \"^$F$!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Charakteristicka r ovnice ma tedy 2 komplexni sdruzene koreny. " }}{PARA 0 "" 0 "" {TEXT -1 71 "Zname-li koreny charakteristicke rovnice muzeme vyjadrit obecne reseni." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "ob_reseni:=y(x) =c[1]*exp(-2*x)*cos(x)+c[2]*exp(-2*x)*sin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*ob_reseniG/-%\"yG6#%\"xG,&*(&%\"cG6#\"\"\"F/-%$expG6 #,$F)!\"#F/-%$cosGF(F/F/*(&F-6#\"\"#F/F0F/-%$sinGF(F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Vypocteme derivaci obecneho reseni." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "derivace:=diff(ob_reseni,x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)derivaceG/-%%diffG6$-%\"yG6#%\" xGF,,**(&%\"cG6#\"\"\"F2-%$expG6#,$F,!\"#F2-%$cosGF+F2F7*(F/F2F3F2-%$s inGF+F2!\"\"**\"\"#F2&F06#F?F2F3F2F;F2F=*(F@F2F3F2F8F2F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Podminku " }{TEXT 287 1 "y" }{TEXT 304 4 " (0) " }{TEXT 303 2 "= " }{TEXT 306 2 "-1" }{TEXT 305 1 " " }{TEXT -1 39 "dosadime do obecneho reseni a podminku " }{TEXT 308 2 "y'" }{TEXT -1 4 "(0) " }{TEXT 307 2 "= " }{TEXT 310 1 "5" }{TEXT 309 1 " " } {TEXT -1 29 "do derivace obecneho reseni. " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 77 "dosazeni1:=-1=subs(x=0,rhs(ob_reseni)); dosazeni2:= 5=subs(x=0,rhs(derivace));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*dosaz eni1G/!\"\",&*(&%\"cG6#\"\"\"F,-%$expG6#\"\"!F,-%$cosGF/F,F,*(&F*6#\" \"#F,F-F,-%$sinGF/F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*dosazeni2 G/\"\"&,**(&%\"cG6#\"\"\"F,-%$expG6#\"\"!F,-%$cosGF/F,!\"#*(F)F,F-F,-% $sinGF/F,!\"\"**\"\"#F,&F*6#F9F,F-F,F5F,F7*(F:F,F-F,F1F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Ze ziskane soustavy rovnic" }{TEXT 283 1 " " }{TEXT -1 19 "vypocteme konstanty" }{TEXT 284 1 " " }{XPPEDIT 18 0 "c[1];" "6#&%\"cG6#\"\"\"" }{TEXT 311 1 " " }{TEXT -1 2 "a " } {XPPEDIT 18 0 "c[2];" "6#&%\"cG6#\"\"#" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "konstanty:=solve(\{dosazeni1,dosaze ni2\},\{c[1],c[2]\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*konstantyG <$/&%\"cG6#\"\"\"!\"\"/&F(6#\"\"#\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Dosazenim konstant do obecneho reseni dostaneme hledane p artikularni reseni." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "part _reseni:=subs(konstanty,ob_reseni);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%,part_reseniG/-%\"yG6#%\"xG,&*&-%$expG6#,$F)!\"#\"\"\"-%$cosGF(F1! \"\"*(\"\"$F1F,F1-%$sinGF(F1F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Overime spravnost vypoctu pomoci prikazu " }{TEXT 312 6 "dsolve" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "part_resen i1:=dsolve(\{rovnice2,y(0)=-1,D(y)(0)=5\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-part_reseni1G/-%\"yG6#%\"xG,&*&-%$expG6#,$F)!\"#\"\" \"-%$cosGF(F1!\"\"*(\"\"$F1F,F1-%$sinGF(F1F1" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 15 "Pomoci prikazu " }{TEXT 288 6 "DEplot" }{TEXT -1 12 " z knihovny " }{TEXT 289 7 "DEtools" }{TEXT -1 60 " zobrazime graf parti kularniho reseni spolu s tecnou v bode " }{XPPEDIT 18 0 "x[0] = 0;" "6 #/&%\"xG6#\"\"!F'" }{TEXT -1 93 ". Rovnici tecny hledame ve smernicove m tvaru, smernice odpovida derivaci v nule, usek na ose " }{TEXT 313 2 "y " }{TEXT -1 26 "je funkcni hodnota v nule." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 86 "tecna:=y(x)=simplify(subs(x=0,diff(rhs(part_re seni),x))*x+subs(x=0,rhs(part_reseni)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&tecnaG/-%\"yG6#%\"xG,&F)\"\"&\"\"\"!\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 331 "graf_tecny:=plot(rhs(tecna),x=-.25..0.25,th ickness=2,color=red): with(D Etools): graf_reseni:=DEplot(rovnice2,y(x),x=-.25..2.5,[[y(0)=-1,D(y)( 0)=5]],stepsize=.05,linecolor=black,thickness=2): \+ with(plots): display([graf_tecny,graf_reseni],tickmarks =[4,3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'C URVESG6%7S7$$!3++++++++D!#=$!3+++++++]A!#<7$$!3mmm;HU,\"R#F*$!37LLe9r] &>#F-7$$!3=L$3FH'='H#F*$!3QmTNYJ4[@F-7$$!3ommTgBa*=#F*$!3IL$3-=rZ4#F-7 $$!3_mm\"H_\">#3#F*$!3;L$e9w&4T?F-7$$!3YL$3_!4Nv>F*$!3omTg_an()>F-7$$! 3qm;/wfHw=F*$!3VL3-))z9Q>F-7$$!3%***\\PM.ttc\"F*$!39+vVB<'4y\"F-7$$!3ELL$epjJX \"F*$!3emm\"z%=eE1d\"F-7$$!3,++]i^)o.\"F*$!3 )****\\7eU%=:F-7$$!3'om;/^?7U*!#>$!3SL3_D51r9F-7$$!3/MLLeaR%H)Fap$!3tm m\"Hx>ZT\"F-7$$!3(\\LLL$o#)RtFap$!3!omm;M\"*pO\"F-7$$!3D,+v$fO%HiFap$! 34+voH=Z68F-7$$!3qNLL$3`lC&Fap$!3#ommTlFBE\"F-7$$!3#*)**\\P4u\"oTFap$! 3$)*\\(o/(3%37F-7$$!3n***\\7G-89$Fap$!3!**\\iS6lq:\"F-7$$!3sKL$3Fp)p?F ap$!3mm;ajM\\.6F-7$$!3eLL3-$ff3\"Fap$!3imT5lzHa5F-7$$!3)Grmm\"z%zY#!#@ $!3OL$eR(RB,5F-7$$\"3CHLe*)>px5Fap$!3QN$3_+a6Y*F*7$$\"3V(**\\Pf4t.#Fap $!3%=+DJ?X8)*)F*7$$\"3qIL$e*GstIFap$!3mML3_&QJY)F*7$$\"3D)*****\\#RW9% Fap$!3)3++]P!yFzF*7$$\"3#[***\\7j#>>&Fap$!3f-+vVo./uF*7$$\"32'**\\i!RU 0iFap$!3'>+vo/)G(*oF*7$$\"3d&***\\(=S2L(Fap$!3A-+D1*HYL'F*7$$\"3;jmm;p )=M)Fap$!3UommTl0HeF*7$$\"3.*****\\(=]@%*Fap$!3/,+]i!\\#*G&F*7$$\"35L$ e*[$z*R5F*$!3ZM$3_D.,![F*7$$\"3#*****\\iC$p9\"F*$!3U++](oP`E%F*7$$\"39 m;H2qcZ7F*$!3')p;aj\\;iPF*7$$\"3q**\\7.\"fFN\"F*$!3_,]P%[/iB$F*7$$\"3Y mm;/Ogb9F*$!3tnm;z>)>s#F*7$$\"3y**\\ilAFj:F*$!3;,](=nQO=#F*7$$\"3YLLL$ )*ppm\"F*$!3ELLL$3]^m\"F*7$$\"3?LL3xe,t#[(>F*$!3m1++]P!*e7Fap7$$\"3im mT&G!e&3#F*$\"3kDL$3F9!zUFap7$$\"3;LLL$)Qk%=#F*$\"3$=lmm;%>K#*Fap7$$\" 37+]iSjE!H#F*$\"3g+]7.F*7$$\"3+ +++++++DF*Ffz-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F`[lF_[l-%*THICKNES SG6#\"\"#-F$6&7Z7$$!#D!\"#$!3px(f#Hq;@GF-7$$!#?F[\\l$!3]Ah5ppA^BF-7$$! #:F[\\l$!3)>e&3'[i)R>F-7$$!#5F[\\l$!35&QaJE66e\"F-7$$!\"&F[\\l$!3M!R\\ `J'\\p7F-7$F_[l$!\"\"F`[l7$$\"\"&F[\\l$!3/mm;/wP!o(F*7$$\"#5F[\\l$!33j W2&z3Vp&F*7$$\"#:F[\\l$!3$z?IO$z!Q+%F*7$$\"#?F[\\l$!3yvuQ6*HWd#F*7$$\" #DF[\\l$!3!Q!4v%>=]P\"F*7$$\"#IF[\\l$!3]1OfCKhuPFap7$$\"#NF[\\l$\"3!RX :7?4aV%Fap7$$\"#SF[\\l$\"327Ond2q56F*7$$\"#XF[\\l$\"35\"H\"p,XNW;F*7$$ \"#]F[\\l$\"37LZstImi?F*7$$\"#bF[\\l$\"3/<&p**z+=Q#F*7$$\"#gF[\\l$\"35 )[dlYKhh#F*7$$\"#lF[\\l$\"3YDHpf(y$yFF*7$$\"#qF[\\l$\"3Qo(HPKq(zGF*7$$ \"#vF[\\l$\"37E&\\ov!>IHF*7$$\"#!)F[\\l$\"3o#>a7$*3$QHF*7$$\"#&)F[\\l$ \"3x$3ta2,<\"HF*7$$\"#!*F[\\l$\"3[op#H4jp&GF*7$$\"#&*F[\\l$\"3s,oW:O\" )zFF*7$$\"$+\"F[\\l$\"37$Q.\\I)=&o#F*7$$\"$0\"F[\\l$\"3]WY8AfKxDF*7$$ \"$5\"F[\\l$\"3!of=9oY)fCF*7$$\"$:\"F[\\l$\"3.Vh5-M#eL#F*7$$\"$?\"F[\\ l$\"3V_k!y&f%y?#F*7$$\"$D\"F[\\l$\"3Ccpm3&z!y?F*7$$\"$I\"F[\\l$\"3+?ij %)pJ[>F*7$$\"$N\"F[\\l$\"3r!*)[yEE+#=F*7$$\"$S\"F[\\l$\"3aB4.,DR%p\"F* 7$$\"$X\"F[\\l$\"3W#>,m#fNs:F*7$$\"$]\"F[\\l$\"3/aF[\\l$\"3a'Q+dvKS2(Fap7$$\"$&>F[\\l$\"3WCF#>Xm/R'Fap7$$\"$+#F[\\l $\"3o-M15a[edFap7$$\"$0#F[\\l$\"3oG%Q]F\"*e<&Fap7$$\"$5#F[\\l$\"3=*yLw CD.k%Fap7$$\"$:#F[\\l$\"3S6d%R'QK\\TFap7$$\"$?#F[\\l$\"3MZ6E[eO+PFap7$ $\"$D#F[\\l$\"3/58BF6\"4H$Fap7$$\"$I#F[\\l$\"3z_7r%)HV=HFap7$$\"$N#F[ \\l$\"3)pARnBW/e#Fap7$$\"$S#F[\\l$\"3oK]Q9#=XF#Fap7$$\"$X#F[\\l$\"3[gv ;e_I)*>Fap7$$\"$]#F[\\l$\"3CX0)4NX&\\ " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "rovnice3:=diff(y(x),x$2)+2*diff(y(x),x)+y(x)=0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%)rovnice3G/,(-%%diffG6$-%\"yG6#%\"xG -%\"$G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2F*F2\"\"!" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 19 "Pouzijeme proceduru" }{TEXT 316 5 " char" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "read \"difproc.m \": char_rovnice:=char(rovnice3,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-char_rovniceG/,(*$)%'lambdaG\"\"#\"\"\"F+*&F*F+F)F+F +F+F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solve(char_ro vnice,lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$!\"\"F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Charakteristicka rovnice ma dvojnasobny k oren. K ziskani obecneho reseni pouzijeme proceduru " }{TEXT 317 3 "ho m" }{TEXT -1 65 ", ktera hleda obecne reseni dane homogenni diferencia lni rovnice." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ob_reseni:= hom(rovnice3,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*ob_reseniG/- %\"yG6#%\"xG,&*&&%\"cG6#\"\"\"F/-%$expG6#,$F)!\"\"F/F/*(&F-6#\"\"#F/F) F/F0F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Do obecneho reseni a \+ jeho derivace dosadime zadane pocatecni podminky." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 86 "dosazeni1:=2=subs(x=0,rhs(ob_reseni)); dosaz eni2:=-1=subs(x=0,rhs(diff(ob_reseni,x)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*dosazeni1G/\"\"#*&&%\"cG6#\"\"\"F+-%$expG6#\"\"!F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*dosazeni2G/!\"\",&*&&%\"cG6#\"\" \"F,-%$expG6#\"\"!F,F&*&&F*6#\"\"#F,F-F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Ze ziskane soustavy rovnic vypocteme konstanty " } {XPPEDIT 18 0 "c[1];" "6#&%\"cG6#\"\"\"" }{TEXT 315 1 " " }{TEXT -1 2 "a " }{XPPEDIT 18 0 "c[2];" "6#&%\"cG6#\"\"#" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "konstanty:=solve(\{dosazeni1 ,dosazeni2\},\{c[1],c[2]\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*kon stantyG<$/&%\"cG6#\"\"\"\"\"#/&F(6#F+F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Hledane partikularni reseni dostaneme dosazenim konstant \+ do obecneho reseni." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "part _reseni:=y(x)=subs(konstanty,rhs(ob_reseni));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,part_reseniG/-%\"yG6#%\"xG,&-%$expG6#,$F)!\"\"\"\"#* &F)\"\"\"F+F2F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Overime, zda n alezene partikularni reseni vyhovuje zadane rovnici." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "simplify(subs(part_reseni,rovnice3));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Pro nalezeni partkularniho reseni muzeme pouzit take proc eduru " }{TEXT 330 3 "hom" }{TEXT -1 57 ", ve ktere jako treti paramet r zadame pocatecni podminky," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "podminky:=[[0,2],[0,-1]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%) podminkyG7$7$\"\"!\"\"#7$F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "part_reseni2:=hom(rovnice3,y(x),podminky);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-part_reseni2G/-%\"yG6#%\"xG,&-%$expG6#,$F )!\"\"\"\"#*&F)\"\"\"F+F2F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "ne bo lze pouzit proceduru " }{TEXT 331 4 "part" }{TEXT -1 58 ", ktera hl eda partikularni reseni primo z obecneho reseni." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "part_reseni3:=part(ob_reseni,podminky);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%-part_reseni3G/-%\"yG6#%\"xG,&-%$exp G6#,$F)!\"\"\"\"#*&F)\"\"\"F+F2F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Zobrazime graf partikularniho reseni." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 45 "plot(rhs(part_reseni),x=-2.5..4,thickness=2);" }} {PARA 13 "" 1 "" {GLPLOT2D 354 266 266 {PLOTDATA 2 "6&-%'CURVESG6$7fn7 $$!3++++++++D!#<$!3]P#F*7$$!35]7`COsl@F*$!3=f7#R'3EX9F*7$$!3'omT&o]S' 4#F*$!3md2$z/^V%y!#=7$$!3&ommT-Fm-#F*$!3'*RR&zux0-#FU7$$!3gm;zz*[o&>F* $\"3+&zhDJ%z`IFU7$$!3-]7GyDS()=F*$\"3wNDp&Q/RV(FU7$$!3VL3xwh&z\"=F*$\" 3?o'\\(=ZD@6F*7$$!3ymT&)oZ=*o\"F*$\"3=*Q>d[%3$o\"F*7$$!3/+voM%\\eb\"F* $\"34L\"\\Yby[5#F*7$$!3!o;a8%Q&zT\"F*$\"3[>2hdH2.CF*7$$!3-+v$4[+0G\"F* $\"332Xx6g2*e#F*7$$!3$oTgFk1)47F*$\"3wC@)z'*)Q\\EF*7$$!3ULLe/G6R6F*$\" 3!>9M'fxS*o#F*7$$!31+vVLc%o2\"F*$\"3j762&[K)4FF*7$$!3rm;Hi%yX,\"F*$\"3 \">D#GX,*zr#F*7$$!3.****\\()G#Qu)FU$\"37'y**)y!\\&)p#F*7$$!3)4++Dc/hL( FU$\"3;YS<2`OPEF*7$$!3))****\\7r]zfFU$\"3sM*e+Tq%\\DF*7$$!3qp;ajmeZZFU $\"3K\\fE$*y,_CF*7$$!3_ML$e49FG$FU$\"3aKm%G808K#F*7$$!3zLLL$)[xT?FU$\" 3E)\\V>t/E?#F*7$$!3CT+v=dn#)f!#>$\"3)*\\h7\")**yf?F*7$$\"39Lmm;*4[z'F] t$\"3Pc`eXC5K>F*7$$\"31)*\\7yOP\"3#FU$\"3oC1wg$=Kz\"F*7$$\"3F,]PMqI;MF U$\"3a;0rdv)Rm\"F*7$$\"3(\\m;z%*p\"4[FU$\"3\\b'*H`JuL:F*7$$\"3ok;H2HD) 3'FU$\"3qg?fEO<>9F*7$$\"3_IL3xm\"zY(FU$\"3mhVC$)>m,8F*7$$\"3MJ$eke**4! *)FU$\"3c>&Raa?T1*FU7$$\"3 m**\\i?/&\\U\"F*$\"3medQKSqP#)FU7$$\"3Q*\\7y50nb\"F*$\"3y!>EQv'e)\\(FU 7$$\"3?**\\PCi*Hq\"F*$\"3czR+#eDXu'FU7$$\"3@mm;*HXW$=F*$\"3,+dW)G)oBhF U7$$\"3/++vV_zu>F*$\"3OrZU$[2m^&FU7$$\"3/Lek`J(>5#F*$\"3ZS_)o$F*$\"37Yz*p%[*R8#FU7$$ \"3;++]-&osJ$F*$\"3o<%***)=,w#>FU7$$\"37m;/rVDhMF*$\"3nz-2DvI9j39;YeB\"FU7$$\"\"%\"\"!$\"3)40CLLQ*)4\"FU-%' COLOURG6&%$RGBG$\"#5!\"\"$Fd]lFd]lF^^l-%+AXESLABELSG6$Q\"x6\"Q!6\"-%*T HICKNESSG6#\"\"#-%%VIEWG6$;$!#DF]^lFb]l%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 9 "P\370\355klad 4" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Urcete partikularni reseni rovnice " }{TEXT 334 7 "y''''' " }{TEXT -1 0 "" }{TEXT 332 7 "- y' = " } {TEXT 336 1 "0" }{TEXT 335 1 " " }{TEXT -1 23 "pro pocatecni podminky \+ " }{TEXT 333 1 "y" }{TEXT 338 3 "(0)" }{TEXT 337 3 " = " }{TEXT 340 1 "0" }{TEXT 339 4 ", y'" }{TEXT 342 3 "(0)" }{TEXT 341 3 " = " }{TEXT 344 1 "1" }{TEXT 343 4 ", y\"" }{TEXT 346 3 "(0)" }{TEXT 345 3 " = " } {TEXT 348 1 "0" }{TEXT 347 5 ", y'\"" }{TEXT 356 3 "(0)" }{TEXT 355 3 " = " }{TEXT 354 1 "1" }{TEXT 353 7 ", y''''" }{TEXT 352 3 "(0)" } {TEXT 351 3 " = " }{TEXT 350 1 "2" }{TEXT 349 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "rovnice4:=diff(y(x),x$5)-diff(y(x),x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)rovnice4G/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F -\"\"&\"\"\"-F(6$F*F-!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "podminky:=[[0,0],[0,1],[0,0],[0,1],[0,2]];" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%)podminkyG7'7$\"\"!F'7$F'\"\"\"F&F(7$F'\"\"#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Jde o homogenni rovnici 5. radu. U rcime koreny charakteristicke rovnice a nalezneme obecne reseni." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "read \"difproc.m\": \+ solve(char(rovnice4,y(x)),lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 '\"\"!!\"\"\"\"\"^#F%^#F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ob_reseni:=hom(rovnice4,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%*ob_reseniG/-%\"yG6#%\"xG,,*&&%\"cG6#\"\"\"F/-%$expG6#,$F)!\"\"F/F/* &&F-6#\"\"#F/-%$sinGF(F/F/&F-6#\"\"$F/*&&F-6#\"\"%F/-F1F(F/F/*&&F-6#\" \"&F/-%$cosGF(F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Pomoci proc edury " }{TEXT 357 5 "part " }{TEXT -1 34 "urcime hledane partikularni reseni" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "part_reseni:=par t(ob_reseni,podminky);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,part_rese niG/-%\"yG6#%\"xG,(!\"#\"\"\"-%$expGF(F,-%$cosGF(F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "a zkontrolujeme, zda je resenim zadane rovnice. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "simplify(subs(part_rese ni,rovnice4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Zobrazime graf nalezeneho partikularniho \+ reseni." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot(rhs(part_re seni),x=-11..2.2,thickness=2,scaling=constrained);" }}{PARA 13 "" 1 " " {GLPLOT2D 354 266 266 {PLOTDATA 2 "6'-%'CURVESG6$7\\o7$$!#6\"\"!$!3- f6J+wb&*>!#<7$$!33++]cxAr5!#;$!3)e2tV?+&zAF-7$$!31+]F\"=$>Y5F1$!3Fq(\\ (GTk3DF-7$$!3(****\\J=R!=5F1$!3h\\.)4#G\"ys#F-7$$!3i++]?c)p*)*F-$!314A Rrf^!*GF-7$$!3G+]70g&fv*F-$!3c(\\5;+-c%HF-7$$!3\"****\\(*QE\\h*F-$!312 $ooY4>)HF-7$$!3K,+D$)Q<%[*F-$!3UOv%[`g\")*HF-7$$!3'4+]nP@MN*F-$!3;u3kI &ot*HF-7$$!3F++vHZ.=#*F-$!31`K)*)R0'yHF-7$$!3e***\\F3[E3*F-$!3[1$f0iG> %HF-7$$!3'3+]Z&Rh-))F-$!3y-\\\")[)oC\"GF-7$$!3[++v**yZB&)F-$!3D3uYBBR? EF-7$$!3o+++d@NO#)F-$!3ioiY(*f&GP#F-7$$!3s++]Y;X$)zF-$!35O[))[vwG@F-7$ $!3))*****4dX()p(F-$!3mZG\\wK$\\%=F-7$$!3-,++l/(GT(F-$!3#o-_+._Cd\"F-7 $$!3\"3++!HoPPrF-$!3Q/2K;oPU8F-7$$!3E++v9A?()oF-$!3'=R&\\)Q=f<\"F-7$$! 37+++,/s*e'F-$!3#QVF^sS_/\"F-7$$!3u++]UtrjkF-$!30P4)oq#p95F-7$$!3Z+++% G9xL'F-$!3Q`Eon*zr***!#=7$$!3a+]PsF9\">'F-$!3aM/&)Qa=-5F-7$$!3i++vg7dW gF-$!3K\"=Bj\"G'f-\"F-7$$!3.,]PP1$[\"fF-$!3)o8SH#)zV1\"F-7$$!3W,++9+4& y&F-$!36,^rJGV=6F-7$$!3)****\\2'zR+bF-$!3u8')3fV'pG\"F-7$$!3Z++D-QIH_F -$!3s/boXBW+:F-7$$!37++]([Xk%\\F-$!3SBJbYg'4w\"F-7$$!3s++vbKp'o%F-$!3X R9e#yvk,#F-7$$!3>,+]#Q:lS%F-$!3yz^$3tG*)G#F-7$$!3N,+v9$*[:TF-$!3;KTf$* HwXDF-7$$!3W,+Dn-:iQF-$!3m_Y,%=:/t#F-7$$!3'4++b:P&)e$F-$!3wcgH,K8uGF-7 $$!3I,+v'f-sW$F-$!34E\"Gt/H=#HF-7$$!3i,++Q!oeI$F-$!3?<#>oup)\\HF-7$$!3 O,+vX(*fnJF-$!3k=!HjHdv&HF-7$$!35,+]`9LHIF-$!3)>JFiPd`%HF-7$$!35,]7r( \\b*GF-$!3o6_23_h9HF-7$$!37,+v)3o++S&=u(>#F-$!350!\\YE?cZ#F-7$$!3m+++0Ns7>F-$!3? w!GFbr$\"3!Qr8K4H) \\GFbr7$$\"3a++]_&[Se&Fbr$\"3ebZg>]$)ffFbr7$$\"30#*****>))HN\")Fbr$\"3 Y%[&e?N%zU*Fbr7$$\"3&*)***\\`>$f5\"F-$\"33ptUG6Mq9F-7$$\"3T)**\\F!fpO7 F-$\"3[(y#y\"4M@x\"F-7$$\"3'y****>&)fuO\"F-$\"3\"40t7#F-7$$\"3t)* \\i&\\\")o]\"F-$\"3oZ^jwu]wDF-7$$\"3g***\\#RJIY;F-$\"3u%41+!3L7JF-7$$ \"3!))****>WZ(z8>F-$\"3Ww(pQ%H$*Q WF-7$$\"3a)\\i&3Q*[)>F-$\"3uJ7'of5f([F-7$$\"33**\\Psefc?F-$\"3irHy()4R _`F-7$$\"3&)*\\(=OzHG@F-$\"3cfD<&e)[reF-7$$\"3;+++++++AF-$\"3>w(y@Q7lV 'F--%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*F]`l-%+AXESLABELSG6$Q\"x6\"Q!6\"-% *THICKNESSG6#\"\"#-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;F($\"#AF\\`l%( DEFAULTG" 1 2 0 1 10 2 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Cur ve 1" }}}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 45 "Nehomogenni rovnice \+ \226 Metoda variace konstant" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "P \370\355klad 5" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Naleznete obecne reseni rovnice " }{TEXT 358 16 "y'' - 2y' + y = " }{XPPEDIT 18 0 "e^x ;" "6#)%\"eG%\"xG" }{TEXT -1 0 "" }{TEXT 359 3 "/ x" }{TEXT -1 1 "." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "rovnice1:=diff(y(x),x$2)-2*diff(y(x),x)+y (x)=exp(x)/x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)rovnice1G/,(-%%dif fG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2!\"\"F*F2*&-%$e xpGF,F2F-F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Jde o nehomogenni \+ diferencialni rovnici 2. radu, kterou budeme resit metodou variace kon stant. " }}{PARA 0 "" 0 "" {TEXT -1 35 "Prislusna homogenni rovnice ma tvar" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "hom_rovnice:=lhs(r ovnice1)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,hom_rovniceG/,(-%%di ffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2!\"\"F*F2\"\"! " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Uzitim procedury " }{TEXT 360 4 "char" }{TEXT -1 62 " spocitame koreny charakteristicke rovnice \+ a pomoci procedury " }{TEXT 361 3 "hom" }{TEXT -1 48 " nalezneme obecn e reseni homogenni rovnice. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "read \"difproc.m\": solve(char(hom_rovnice,y(x) ),lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "ob_res_hom:=hom(hom_rovnice,y(x)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+ob_res_homG/-%\"yG6#%\"xG,&*&&% \"cG6#\"\"\"F/-%$expGF(F/F/*(&F-6#\"\"#F/F)F/F0F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Partikularni reseni nehomogenni rovnice budeme \+ hledat ve tvaru" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "tvar:=su bs(c[1]=K[1](x),c[2]=K[2](x),ob_res_hom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tvarG/-%\"yG6#%\"xG,&*&-&%\"KG6#\"\"\"F(F0-%$expGF(F 0F0*(-&F.6#\"\"#F(F0F)F0F1F0F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "ktery jsme dostali nahrazenim konstant " }{XPPEDIT 18 0 "c[1];" "6#&% \"cG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[2];" "6#&%\"cG6#\"\"# " }{TEXT -1 10 " funkcemi " }{XPPEDIT 18 0 "K[1](x);" "6#-&%\"KG6#\"\" \"6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "K[2](x);" "6#-&%\"KG6#\"\" #6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 25 "Vypocteme prv ni derivaci." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "der1:=diff( tvar,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%der1G/-%%diffG6$-%\"yG6 #%\"xGF,,,*&-F'6$-&%\"KG6#\"\"\"F+F,F5-%$expGF+F5F5*&F1F5F6F5F5*(-F'6$ -&F36#\"\"#F+F,F5F,F5F6F5F5*&F " 0 "" {MPLTEXT 1 0 45 "podminka1:=op(1,rhs(der1))+op(3,rhs(der1))=0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%*podminka1G/,&*&-%%diffG6$-&%\"KG6# \"\"\"6#%\"xGF1F/-%$expGF0F/F/*(-F)6$-&F-6#\"\"#F0F1F/F1F/F2F/F/\"\"! " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Dosazenim dostavame:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "der1upr:=lhs(der1)=rhs(der1) -lhs(podminka1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(der1uprG/-%%dif fG6$-%\"yG6#%\"xGF,,(*&-&%\"KG6#\"\"\"F+F3-%$expGF+F3F3*&-&F16#\"\"#F+ F3F4F3F3*(F7F3F,F3F4F3F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "der2:=diff(der1upr,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%der2G/-% %diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#,.*&-F'6$-&%\"KG6#\"\"\"F+F,F9-%$e xpGF+F9F9*&F5F9F:F9F9*&-F'6$-&F76#F0F+F,F9F:F9F9*(F0F9F@F9F:F9F9*(F>F9 F,F9F:F9F9*(F@F9F,F9F:F9F9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Dos azenim do puvodni rovnice mame:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "podminka2:=simplify(subs(diff(y(x),x$2)=rhs(der2),y(x)=rhs(tva r),rovnice1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*podminka2G/,(*&-% %diffG6$-&%\"KG6#\"\"\"6#%\"xGF1F/-%$expGF0F/!\"\"*&-F)6$-&F-6#\"\"#F0 F1F/F2F/F/*(F6F/F1F/F2F/F4*&F2F/F1F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "V ziskane soustave rovnic zavedeme pro jednoduchost substituci \+ - prvni derivace hledanych funkci " }{XPPEDIT 18 0 "K[1](x);" "6#-&%\" KG6#\"\"\"6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "K[2](x);" "6#-&%\" KG6#\"\"#6#%\"xG" }{TEXT -1 10 " oznacime " }{TEXT 364 4 "A, B" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "soustava:= subs(\{diff(K[1](x),x)=A,diff(K[2](x),x)=B\},\{podminka1,podminka2\}); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)soustavaG<$/,&*&%\"AG\"\"\"-%$e xpG6#%\"xGF*F**(%\"BGF*F.F*F+F*F*\"\"!/,(F(!\"\"*&F0F*F+F*F*F/F4*&F+F* F.F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Vypocteme prvni derivace \+ funkci " }{XPPEDIT 18 0 "K[1](x);" "6#-&%\"KG6#\"\"\"6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "K[2](x);" "6#-&%\"KG6#\"\"#6#%\"xG" }{TEXT -1 28 " a ulozime je do promennych " }{TEXT 365 4 "A, B" }{TEXT -1 2 " . " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "konst_der:=solve(sous tava,\{A,B\}); assign(konst_der);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %*konst_derG<$/%\"BG*&\"\"\"F)%\"xG!\"\"/%\"AGF+" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 10 "Integraci " }{TEXT 366 2 "A " }{TEXT -1 2 "a " } {TEXT 367 1 "B" }{TEXT -1 51 " dostaneme vzhledem ke stanovene substit uci funkce " }{XPPEDIT 18 0 "K[1](x);" "6#-&%\"KG6#\"\"\"6#%\"xG" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "K[2](x);" "6#-&%\"KG6#\"\"#6#%\"xG" } {TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "konst1:=i nt(A,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'konst1G,$%\"xG!\"\"" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "konst2:=int(B,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'konst2G-%#lnG6#%\"xG" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 39 "Hledane partikularni reseni pak ma tvar" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "part_reseni:=simplify(subs(K [1](x)=konst1,K[2](x)=konst2,tvar));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%,part_reseniG/-%\"yG6#%\"xG,&*&F)\"\"\"-%$expGF(F,!\"\"*(-%#lnGF(F ,F)F,F-F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "Obecne reseni neh omogenni rovnice je souctem obecneho reseni prislusne homogenni rovnic e a partikularniho reseni nehomogenni rovnice. " }}{PARA 0 "" 0 "" {TEXT -1 231 "Jelikoz Maple pri integraci, jejimz vysledkem je funkce \+ obsahujici prirozeny logaritmus, nezohlednuje definicni obor (tj. neuz avira argument prirozeneho logaritmu do absolutni hodnoty), vyresime t uto odlisnost rucnim prepsanim. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "ob_reseni:=y(x)=rhs(ob_res_hom)-x*exp(x)+ln(abs(x))*x *exp(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*ob_reseniG/-%\"yG6#%\"x G,**&&%\"cG6#\"\"\"F/-%$expGF(F/F/*(&F-6#\"\"#F/F)F/F0F/F/*&F)F/F0F/! \"\"*(-%#lnG6#-%$absGF(F/F)F/F0F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Reseni ziskane prikazem " }{TEXT 363 6 "dsolve" }{TEXT -1 37 " \+ bude mit take tuto odlisnost. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ob_dsolve:=dsolve(rovnice1,y(x));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%*ob_dsolveG/-%\"yG6#%\"xG,**&F)\"\"\"-%$expGF(F,!\" \"*(-%#lnGF(F,F)F,F-F,F,*&%$_C1GF,F-F,F,*(%$_C2GF,F)F,F-F,F," }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Nyni zobrazime grafy vybranych par tikularnich reseni pro pevnou hodnotu konstanty " }{XPPEDIT 18 0 "c[1] = 1;" "6#/&%\"cG6#\"\"\"F'" }{TEXT -1 15 ", za konstantu " }{XPPEDIT 18 0 "c[2];" "6#&%\"cG6#\"\"#" }{TEXT -1 36 " postupne volime hodnoty \+ 1, 0, a -1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 299 "with(plots) : g1:=plot(subs(c[1]=1,c[2]=1,rhs(ob_reseni)),x=-2.5..1.1,color=red): \+ g2:=plot(subs(c[1]=1,c[2]=0,rhs(ob_reseni)),x=-2.5..1.1,color=blue): g 3:=plot(subs(c[1]=1,c[2]=-1,rhs(ob_reseni)),x=-2.5..1.1,color=black): \+ display(g1,g2,g3,thickness=2,tickmarks =[4,3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 354 266 266 {PLOTDATA 2 "6)-%'C URVESG6$7jn7$$!3++++++++D!#<$!3/VbQ+J\\f5!#=7$$!3-+++X-`@CF*$!3@%yF3Zx N,\"F-7$$!3!****\\28aKN#F*$!3(>P0QE'=Q'*!#>7$$!3!)****\\*\\qkF#F*$!3'3 h*=#F*$!3TL(z?)*f+8)F87$$!3A++v^EDA@F*$!3;H!yH) *[\"\\rF87$$!3=++v-J$40#F*$!3eoC,)zGd3'F87$$!31++vSe3x>F*$!33ru@Jdp8[F 87$$!3*****\\n*Gr+>F*$!3?znxIA_)H$F87$$!3*)***\\<)[eC=F*$!35hh'fR()zc \"F87$$!3#)*****4'yFY.'y=]F87$$!3;+++X>p@:F*$\"3wD65X Ei&)yF87$$!3!******pJdlW\"F*$\"3%ytse'4v'4\"F-7$$!3-++vwyKy8F*$\"3dLQ( 3V)[09F-7$$!31+++tk>(H\"F*$\"3.^pd'e*[5=F-7$$!3.+++KvYG7F*$\"35+yKl?U( =#F-7$$!33++vM%>&[6F*$\"3S&z3U*[tmEF-7$$!3?+++A=vx5F*$\"37!3xELJ*GJF-7 $$!31++vM&3,+\"F*$\"3q,j)4!f*zn$F-7$$!3k)***\\Uwth#*F-$\"3x++p'3%*>C%F -7$$!3K*****\\(eI!\\)F-$\"3_ahyrSxs[F-7$$!3=++](p!*=y(F-$\"3?QXO1+h)[& F-7$$!3A,++D#px,(F-$\"3CW-w&z%**)='F-7$$!31-+]n<1CiF-$\"3#*e3<=3H]pF-7 $$!3H-+]#4PJ`&F-$\"3fgOt\"[&[LwF-7$$!3-,++:&>py%F-$\"31)Q@psP4Q)F-7$$! 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Pomoci volitelneho parametru " }{TEXT 373 6 "\"fund\"" }{TEXT -1 103 " nechame zobrazit take fundamentalni system zadane rovn ice, ktery budeme potrebovat pri dalsim vypoctu." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "ob_res_hom:=hom(lhs(rovnice2)=0,y(x),\"fund\" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$-%$sinG6#,$%\"xG\"\"#-%$cosGF& " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+ob_res_homG/-%\"yG6#%\"xG,&*&&% \"cG6#\"\"\"F/-%$sinG6#,$F)\"\"#F/F/*&&F-6#F4F/-%$cosGF2F/F/" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Partikularni reseni nehomogenni ro vnice budeme hledat v nasledujicim tvaru." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 59 "tvar:=y(x)=subs(c[1]=K[1](x),c[2]=K[2](x),rhs(ob_re s_hom));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tvarG/-%\"yG6#%\"xG,&*& -&%\"KG6#\"\"\"F(F0-%$sinG6#,$F)\"\"#F0F0*&-&F.6#F5F(F0-%$cosGF3F0F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "Tak, jak je v praktickem vypoc tu obvykle, vynechame odvozeni a prejdeme rovnou k soustave rovnic pro prvni derivace neznamych funkci " }{XPPEDIT 18 0 "K[1](x);" "6#-&%\"K G6#\"\"\"6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "K[2](x);" "6#-&%\"K G6#\"\"#6#%\"xG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "podminka1:=diff(K[1](x),x)*fund[1]+diff(K[2](x),x)*fund[2]=0; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*podminka1G/,&*&-%%diffG6$-&%\"K G6#\"\"\"6#%\"xGF1F/-%$sinG6#,$F1\"\"#F/F/*&-F)6$-&F-6#F6F0F1F/-%$cosG F4F/F/\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "podminka2:=d iff(K[1](x),x)*diff(fund[1],x)+diff(K[2](x),x)*diff(fund[2],x)=rhs(rov nice2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*podminka2G/,&*&-%%diffG6 $-&%\"KG6#\"\"\"6#%\"xGF1F/-%$cosG6#,$F1\"\"#F/F6*(F6F/-F)6$-&F-6#F6F0 F1F/-%$sinGF4F/!\"\"*&F/F/F=F?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Vyresime danou soustavu, ve ktere pro jednoduchost oznacime prvni der ivace hledanych funkci " }{XPPEDIT 18 0 "K[1](x);" "6#-&%\"KG6#\"\"\"6 #%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "K[2](x);" "6#-&%\"KG6#\"\"#6# %\"xG" }{TEXT -1 9 " symboly " }{TEXT 372 4 "A, B" }{TEXT -1 2 ". " }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "konst_der:=solve(subs(diff (K[1](x),x)=A,diff(K[2](x),x)=B,\{podminka1,podminka2\}),\{A,B\}); ass ign(konst_der);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*konst_derG<$/%\" BG#!\"\"\"\"#/%\"AG,$*&-%$cosG6#,$%\"xGF*\"\"\"-%$sinGF1F)#F4F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Intregraci dostaneme hledane funkc e " }{XPPEDIT 18 0 "K[1](x);" "6#-&%\"KG6#\"\"\"6#%\"xG" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "K[2](x);" "6#-&%\"KG6#\"\"#6#%\"xG" }{TEXT -1 1 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "konst1:=int(A,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'konst1G,$-%#lnG6#-%$sinG6#,$%\"xG\" \"##\"\"\"\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "konst2:= int(B,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'konst2G,$%\"xG#!\"\"\" \"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Upravime vyjadreni prvni k onstanty uzavrenim argumentu logaritmu do absolutni hodnoty." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "konst1upr:=1/4*ln(abs(sin(2* x)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*konst1uprG,$-%#lnG6#-%$abs G6#-%$sinG6#,$%\"xG\"\"##\"\"\"\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Vyjadrime obecne reseni nehomogenni rovnice." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "ob_reseni:=y(x)=rhs(ob_res_hom)+subs(K[1] (x)=konst1upr,K[2](x)=konst2,rhs(tvar));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*ob_reseniG/-%\"yG6#%\"xG,**&&%\"cG6#\"\"\"F/-%$sinG6#,$F)\"\" #F/F/*&&F-6#F4F/-%$cosGF2F/F/*(#F/\"\"%F/-%#lnG6#-%$absG6#F0F/F0F/F/*& #F/F4F/*&F)F/F8F/F/!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Vysle dek muzeme overit pouzitim procedury " }{TEXT 371 8 "varconst" }{TEXT -1 126 ", ktera automatizuje predchozi postup pomoci Mapleovskeho prog ramovaciho jazyka.Rozdil je pouze v pridane absolutni hodnote. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "varconst(rovnice2,y(x)); " } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(overeniG/-%\"yG6#%\"xG,**&&%\"cG6# \"\"\"F/-%$sinG6#,$F)\"\"#F/F/*&&F-6#F4F/-%$cosGF2F/F/*(#F/\"\"%F/F0F/ -%#lnG6#F0F/F/*&#F/F4F/*&F8F/F)F/F/!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Vykreslime nektera partikularni reseni pro pevnou hodnotu konstanty " }{XPPEDIT 18 0 "c[1] = 0;" "6#/&%\"cG6#\"\"\"\"\"!" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 296 "with(plot s): g1:=plot(subs(c[1]=0,c[2]=-1,rhs(ob_reseni)),x=-2.5..2.5,numpoints =200,color=red): g2:=plot(subs(c[1]=0,c[2]=0,rhs(ob_reseni)),x=-2.5..2 .5,numpoints=200,color=blue): g3:=plot(subs(c[1]=0,c[2]=1,rhs(ob_resen i)),x=-2.5..2.5,numpoints=200,color=black): display(g1,g2,g3,thickness =2); " }}{PARA 13 "" 1 "" {GLPLOT2D 354 266 266 {PLOTDATA 2 "6(-%'CU RVESG6$7[x7$$!3++++++++D!#<$\"37$Hi\"Rl/'3'!#>7$$!3orV([(>rtCF*$\"3+?I HbE-Q[F-7$$!3CpQx/*Q3X#F*$\"3eyK!)=D(*)z$F-7$$!3\\IhAXd6DCF*$\"3_)*o)= [yYo#F-7$$!3%pRze2A#*R#F*$\"3H7iD*o'HB;F-7$$!3o:Jiu9XtBF*$\"3!QXTaBCXH '!#?7$$!3'*f>RG)e&\\BF*$!3wbyh'y+:M#FG7$$!3Vyc8x$>[K#F*$!3Y`MMp5hn5F-7 $$!3S\"Gc7!QB*H#F*$!3u5=!>E'Hi=F-7$$!33;Kky-ttAF*$!3m:FK$[5Se#F-7$$!3Y rU&3x'\\ZAF*$!3!HH;wj\\0D$F-7$$!3vsX\"HG!RCAF*$!3MiyUcA$>x$F-7$$!3Ic7D ]yP)>#F*$!3-31DA())HG%F-7$$!35,-/3'eA<#F*$!3q(z\"z(ebGr%F-7$$!3ef>Ryy3 Z@F*$!31dKKKF*$!3K;(e0[5\"\\_ F-7$$!3O(\\**)H]vs>F*$!3mH$o@;![m\\F-7$$!3u*)zf>9\"p%>F*$!3qQ(erE&onXF -7$$!3\"e>Ry\"*yJ#>F*$!3#\\\"\\<$)Go/TF-7$$!3e8Fa3-e(*=F*$!3Yg@`[>G(\\ $F-7$$!3?\\)pRM!*4(=F*$!3=([T-d!QUFF-7$$!3KnMp)yVy%=F*$!3GE^oM*3s(>F-7 $$!3vrV([([%G#=F*$!3)>eYN^z8.\"F-7$$!3#)oPv]&=qz\"F*$\"3Qy4%\\KvSY)!#@ 7$$!3#*yd:JDvretdp\"F*$\"33U$RV\\:,5'F-7$$!39 _/4=Etp;F*$\"31>R_qet'>)F-7$$!3a5@UMc8Y;F*$\"3d)z%=])4w.\"!#=7$$!3NiC \\)*yL?;F*$\"3#zblwb\")oJ\"Fdv7$$!3=OsWRU1'f\"F*$\"3#)*>n%3V2S;Fdv7$$! 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38 "Naleznete partikularni reseni rovnice " }{TEXT 374 33 "y' '' + 2y'' + 4y' + 8y = 32cos2x " }{TEXT -1 23 "pro pocatecni podminky \+ " }{TEXT 375 25 "y(0)=2, y'(0)=6, y''(0)=0" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 76 "rovnice3:=diff(y(x),x$3)+2*diff(y(x),x$2)+4*diff(y( x),x)+8*y(x)=32*cos(2*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)rovnic e3G/,*-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"$\"\"\"*&\"\"#F2-F(6$F*-F/6 $F-F4F2F2*&\"\"%F2-F(6$F*F-F2F2*&\"\")F2F*F2F2,$-%$cosG6#,$F-F4\"#K" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "podminky:=[[0,2],[0,6],[0, 0]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)podminkyG7%7$\"\"!\"\"#7$F' \"\"'7$F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "Pro informaci vyp octeme koreny charakteristicke rovnice a vyjadrime obecne reseni prisl usne homogenni rovnice." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " read \"difproc.m\": solve(char(lhs(rovnice3)=0,y(x)),lambda);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%!\"#^#\"\"#^#F#" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "hom(lhs(rovnice3)=0,y(x));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&&%\"cG6#\"\"\"F--%$expG6#,$F'!\"#F- F-*&&F+6#\"\"#F--%$sinG6#,$F'F6F-F-*&&F+6#\"\"$F--%$cosGF9F-F-" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Pro nalezeni obecneho reseni nehom ogenni rovnice pouzijeme proceduru " }{TEXT 376 8 "varconst" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ob_reseni:=varco nst(rovnice3,y(x)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*ob_reseniG/ -%\"yG6#%\"xG,0*&&%\"cG6#\"\"\"F/-%$expG6#,$F)!\"#F/F/*&&F-6#\"\"#F/-% $sinG6#,$F)F8F/F/*&&F-6#\"\"$F/-%$cosGF;F/F/*&F8F/FAF/F/F9F/*(F8F/F9F/ F)F/F/*(F8F/FAF/F)F/!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Do o becneho reseni a jeho prvni a druhe derivace postupne dosadime zadane \+ pocatecni podminky." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "dosa zeni1:=2=subs(x=0,rhs(ob_reseni));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%*dosazeni1G/\"\"#,,*&&%\"cG6#\"\"\"F,-%$expG6#\"\"!F,F,*&&F*6#F&F,-% $sinGF/F,F,*&&F*6#\"\"$F,-%$cosGF/F,F,F4F,*&F&F,F:F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "dosazeni2:=6=subs(x=0,diff(rhs(ob_r eseni),x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*dosazeni2G/\"\"',**& &%\"cG6#\"\"\"F,-%$expG6#\"\"!F,!\"#*(\"\"#F,&F*6#F3F,-%$cosGF/F,F,*(F 3F,&F*6#\"\"$F,-%$sinGF/F,!\"\"*&F3F,F" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "dosazeni3:=0=subs(x=0,diff(rhs(ob_reseni),x$2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*dosazeni3G/\"\"!,**&&%\"cG6#\"\" \"F,-%$expG6#F&F,\"\"%*(F0F,&F*6#\"\"#F,-%$sinGF/F,!\"\"*(F0F,&F*6#\" \"$F,-%$cosGF/F,F7*&F0F,F5F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Resenim soustavy dostaneme hodnoty neznamych konstant " }{XPPEDIT 18 0 "c[1];" "6#&%\"cG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[2];" " 6#&%\"cG6#\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[3];" "6#&%\"cG6#\" \"$" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "kons tanty:=solve(\{dosazeni1,dosazeni2,dosazeni3\},\{c[1],c[2],c[3]\});" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*konstantyG<%/&%\"cG6#\"\"\"\"\"!/& F(6#\"\"#\"\"$/&F(6#F0F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "Dosa zenim do obecneho reseni dostaneme partikularni reseni vyhovujici zada nym pocatecnim podminkam. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "part_reseni:=y[p](x)=subs(konstanty,rhs(ob_reseni));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,part_reseniG/-&%\"yG6#%\"pG6#%\"xG,*-%$si nG6#,$F,\"\"#\"\"%*&F2\"\"\"-%$cosGF0F5F5*(F2F5F.F5F,F5F5*(F2F5F,F5F6F 5!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 " Partikularni reseni mu zeme hledat take pomoci procedury " }{TEXT 378 4 "part" }{TEXT -1 32 " , nebo primo pouzitim procedury " }{TEXT 377 8 "varconst" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "part(ob_reseni,podmin ky);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,*-%$sinG6#,$F' \"\"#\"\"%*&F-\"\"\"-%$cosGF+F0F0*(F-F0F)F0F'F0F0*(F-F0F1F0F'F0!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "varconst(rovnice3,y(x),po dminky);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,*-%$sinG6#, $F'\"\"#\"\"%*&F-\"\"\"-%$cosGF+F0F0*(F-F0F)F0F'F0F0*(F-F0F1F0F'F0!\" \"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Dosazenim overime, zda toto reseni vyhovuje zadane rovnici." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "subs(y(x)=rhs(part_reseni),rovnice3): simplify(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$-%$cosG6#,$%\"xG\"\"#\"#KF$" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Zobrazime graf nalezeneho partikul arniho reseni." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "plot(rhs( part_reseni),x=-8..8,thickness=2,color=black);" }}{PARA 13 "" 1 "" {GLPLOT2D 354 266 266 {PLOTDATA 2 "6'-%'CURVESG6#7_u7$$!\")\"\"!$!3%3< !eW5Fp?!#;7$$!3cLLL$Q6G\"z!#<$!3=(\\9`pc!4>F-7$$!37nmmmFiDyF1$!3Y$QN\" )RQip\"F-7$$!3z******\\TVQxF1$!38X\"f?Tc$Q9F-7$$!3OLLLLbC^wF1$!3YC%p*o T7W6F-7$$!3Mmm;%=$QvvF1$!3a8\"eiuE*f')F17$$!3A+++N3_*\\(F1$!3GHk&G]>Gu &F17$$!3@LL$e[eOU(F1$!3%\\hl78O%eFF17$$!3?mmmOhzZtF1$\"3Ysb%*F17$$!3LLLL`b`1qF1$\"3g5w!G&H\"=>\" F-7$$!3oKLL8UxMoF1$\"3Q^GrI+Zp:F-7$$!3#HLLL(G,jmF1$\"3C\"z+lM1Nv\"F-7$ $!3\"****\\iiw-i'F1$\"3!3<%[o%ouw\"F-7$$!3!pmm\"z.axlF1$\"3Gj`Yz^vov^;F-7$$!30nmm'*G7@jF1$\"3!Q[GS4Ab_\"F-7$$!3o+ ++5]jihF1$\"3_uz&y^3B>\"F-7$$!3XLLLBr9/gF1$\"3Qe)=v;b?k(F17$$!3E+++5Y4 AfF1$\"3xlozvE,._F17$$!3=mmm'4U+%eF1$\"3Uoo\">yU)*p#F17$$!3+LLL$e*)zv& F1$\"3'QOQ[PFW7$$!3!)******pq$fn&F1$!3U+Ntz#zGB#F17$$!3YLLLV#y5f&F 1$!3LUFL!*H>7YF17$$!39nmm;%>i]&F1$!3Ky/&[<.?z'F17$$!3\"3+++fg8U&F1$!3M )fTnSa$=()F17$$!3fLLLj<]O`F1$!3hIAzQ\\iM5F-7$$!3')******z]\">D&F1$!3y$ R2a^0P;\"F-7$$!3)pmmmRGt;&F1$!3+&pk!>(GsD\"F-7$$!3c+++b].D^F1$!3M,V!o^ &>!H\"F-7$$!39MLL8(f&QJ\"F-7$$!3#omm;P[//&F1$!3?#4]3#=@ G8F-7$$!3Q+++I]:)*\\F1$!3Dgiy(oILL\"F-7$$!3UMLeH6la\\F1$!3`v`xH!)4H8F- 7$$!3dnm;Hs96\\F1$!3sE=_JPW:8F-7$$!3s++vGLkn[F1$!3AV!pN=YEH\"F-7$$!3(Q LL$G%RT#[F1$!3)*\\31Jo/h7F-7$$!3<++]F;8PZF1$!3u@$QY?+J<\"F-7$$!3YmmmEQ 7]YF1$!3MS4d-.Wb5F-7$$!3))*******y]o\\%F1$!3gA%pfUY8!zF17$$!3HLLL`xdVV F1$!3!>ljo=&)pv%F17$$!3!ommmhG5<%F1$!32_SZ#H0Y,\"F17$$!3I+++![z%)*RF1$ \"3#Qz%4/5qJDF17$$!3@+++S=ADQF1$\"3Iwyq&)pyDbF17$$!35++++U'>l$F1$\"3q& R;Y$[0!o(F17$$!3K+++!G\"[oNF1$\"3w`$4Dc6)o$)F17$$!33+++g$)*\\[$F1$\"3 \"3Zp9*\\W>))F17$$!3t********oDVMF1$\"3K5k@/KQb*)F17$$!3$)******Ra^,MF 1$\"3Q7'oQ74D.*F17$$!3%*******zRxfLF1$\"3gtO`>y)=0*F17$$!3/+++?D.=LF1$ \"3JL)Quf]\\,*F17$$!3smmmT:TmJF1$\"37Lz#o83$Q%)F17$$!3SLLLj0z9IF1$\"3= Ec6#zRNE(F17$$!39+++b&)\\MGF1$\"3ZjjGsze0`F17$$!3!pmmma1Ul#F1$\"359;oa q\")>IF17$$!3/nmmmbZ,DF1$\"3ccN\\$3r`/\"F17$$!3=nmm'eW([BF1$!3!f[()3c$ *>(zFW7$$!3yLLL[@3r@F1$!3;r[^%HMPh#F17$$!3S+++5(>M*>F1$!3%3kW1e,F'RF17 $$!3;nm;H!*y9>F1$!3OD3lt/H&R%F17$$!3\"RLL$[$eh$=F1$!3^&eWX%zH!)4&F17$$!3gmm;Hg3?9F1$! 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3yM:I]/C)o\"F-7$$\"3q****\\<$)\\(3'F1$!3k6\"3t$=dd:F-7$$\"3KmmmJ*3[;'F 1$!3CF28fO:(Q\"F-7$$\"3&HLLea>@C'F1$!3GOns%=D,=\"F-7$$\"3d******f,V>jF 1$!3\"4yNG@IeS*F17$$\"3ULLL[o.3kF1$!3#*[\"G@(p4DjF17$$\"3QmmmONk'\\'F1 $!3YV_zAGEqHF17$$\"3M*****\\A]_e'F1$\"3tN7E(>;Ng&FW7$$\"3?LLL8p&Qn'F1$ \"3W21#)3\"G.;%F17$$\"3Ummm\"z2Jv'F1$\"3sAyw@n,[tF17$$\"3j******p'eB$o F1$\"3B*o#\\F29U5F-7$$\"3&GLL$[&4;\"pF1$\"3Tz\"[AF-7$$\"3Q++]d**GptF1$\"3c=SuOV#eE#F-7$$\"3P+++Dws4uF1$ \"3cLP2!3&**oAF-7$$\"3\"3+](ek%*HuF1$\"3!y2+IF'4lAF-7$$\"3O++]#Hl,X(F1 $\"3#*=&eL.BvD#F-7$$\"3#****\\i7%QquF1$\"3U#z?Q1piC#F-7$$\"3O+++gHg!\\ (F1$\"38](y\"oSLJAF-7$$\"3O+++&Hy9d(F1$\"3lw!*4,%*)\\8#F-7$$\"3O+++ION _wF1$\"3Y`:'))Q(H\")>F-7$$\"3F++]A_ERxF1$\"3?tTaCumaBZ\"F-7$$\"34++]2%)38zF1$\"3aB*36\\T\"R6F-7$$\"\")F*$\"3A zgA\"Rm\"\\wF1-%+AXESLABELSG6$Q\"x6\"Q!6\"-%'COLOURG6&%$RGBGF*F*F*-%*T HICKNESSG6#\"\"#-%%VIEWG6$;F(Fb^n%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 51 "Nehomogenni rovnice \226 Metoda neurcitych koeficientu " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "P\370\355klad 8" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 379 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Naleznete obecne reseni nehomogenni rovnice " }{TEXT 380 12 "y' ' - 9y = 9" }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 39 "rovnice1:=diff(y(x),x$2)-9*y(x)=9*x^2; " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%)rovnice1G/,&-%%diffG6$-%\"yG6#%\"xG -%\"$G6$F-\"\"#\"\"\"*&\"\"*F2F*F2!\"\",$*$)F-F1F2F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "Vypocteme koreny charakteristicke rovnice a nal ezneme obecne reseni prislusne homogenni rovnice. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read \"difproc.m\":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "solve(char(lhs(rovnice1)=0,y(x)),lambda);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"$!\"$" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 38 "ob_res_hom:=hom(lhs(rovnice1)=0,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+ob_res_homG/-%\"yG6#%\"xG,&*&&%\"cG6#\"\"\"F /-%$expG6#,$F)!\"$F/F/*&&F-6#\"\"#F/-F16#,$F)\"\"$F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 159 "Prava strana zadane rovnice je typu 1. C islo 0 neni korenem charakteristicke rovnice, takze partikularni resen i budeme hledat ve tvaru polynomu druheho stupne." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 31 "part_reseni:=x^0*(A*x^2+B*x+C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,part_reseniG,(*&%\"AG\"\"\")%\"xG\"\"#F(F(*& F*F(%\"BGF(F(%\"CGF(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Tvar hled aneho partikularniho reseni dosadime do zadane nehomogenni rovnice a z jednodusime ji." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "dosazeni :=simplify(subs(y(x)=part_reseni,rovnice1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)dosazeniG/,*%\"AG\"\"#*(\"\"*\"\"\"F'F+)%\"xGF(F+!\" \"*(F*F+F-F+%\"BGF+F.*&F*F+%\"CGF+F.,$*$F,F+F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "Vysledna rovnice obsahuje tri nezname koeficienty. P ostupnym dosazenim hodnot 1,2,3 za promennou " }{TEXT 381 1 "x" } {TEXT -1 40 " dostaneme tri rovnice pro koeficienty " }{TEXT 382 5 "A ,B,C" }{TEXT -1 36 ". Ziskanou soustavu rovnic vyresime." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "koef:=solve(\{seq(subs(x=i,dosazeni ),i=1..3)\},\{A,B,C\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%koefG<%/ %\"AG!\"\"/%\"BG\"\"!/%\"CG#!\"#\"\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "Dosazenim vypoctenych koeficientu do tvaru partikularnih o reseni dostaneme jedno konkretni partikularni reseni." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "part_reseni:=subs(koef,part_reseni) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,part_reseniG,&#!\"#\"\"*\"\"\" *$)%\"xG\"\"#F)!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "Obecne r eseni nehomogenni rovnice je souctem obecneho reseni prislusne homogen ni rovnice a libovolneho partikularniho reseni nehomogenni rovnice." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "ob_reseni:=y(x)=rhs(ob_res _hom)+part_reseni;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*ob_reseniG/-% \"yG6#%\"xG,**&&%\"cG6#\"\"\"F/-%$expG6#,$F)!\"$F/F/*&&F-6#\"\"#F/-F16 #,$F)\"\"$F/F/#F8\"\"*!\"\"*$)F)F8F/F?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Dosazenim overime, ze nalezene reseni vyhovuje zadane rov nici." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "overeni:=simplify( subs(ob_reseni,rovnice1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(overe niG/,$*$)%\"xG\"\"#\"\"\"\"\"*F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Zobrazime grafy nekolika partikularnich reseni." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 244 "with(plots): grafy1:=seq(plot(subs(c[1]= i,c[2]=1,rhs(ob_reseni)),x=-0.45 .. 0.45),i=-1..1): grafy2:=seq(plot(s ubs(c[1]=1,c[2]=i,rhs(ob_reseni)),x=-0.45 .. 0.45),i=-1..0): \+ display(grafy1,grafy2);" }} {PARA 13 "" 1 "" {GLPLOT2D 354 266 266 {PLOTDATA 2 "6)-%'CURVESG6$7S7$ $!35+++++++X!#=$!3![ItA\\2H-%!#<7$$!3=++]7c#QI%F*$!3slI-qXXpPF-7$$!3T+ ](oKNJ8%F*$!3e:i\"34d!fNF-7$$!3;++v[iK(Q^X-\"HF-7$$!3++](ovKtP$F*$! 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Polynom " }{TEXT 390 2 "x " }{TEXT -1 60 "je st upne 1, tedy partikularni reseni budeme hledat ve tvaru" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "part_reseni:=x^1*((A*x+B)*cos(3*x)+ (C*x+D)*sin(3*x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,part_reseniG* &%\"xG\"\"\",&*&,&*&%\"AGF'F&F'F'%\"BGF'F'-%$cosG6#,$F&\"\"$F'F'*&,&*& %\"CGF'F&F'F'%\"DGF'F'-%$sinGF0F'F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Tvar hledaneho partikularniho reseni dosadime do zadane n ehomogenni rovnice. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "dos azeni:=simplify(subs(y(x)=part_reseni,rovnice2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)dosazeniG/,.*(-%$sinG6#,$%\"xG\"\"$\"\"\"%\"AGF.F,F. !#7*(\"\"#F.-%$cosGF*F.F/F.F.*(\"\"'F.F(F.%\"BGF.!\"\"**\"#7F.F3F.%\"C GF.F,F.F.*(F2F.F(F.F;F.F.*(F6F.F3F.%\"DGF.F.*&F,F.F(F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Pro zjisteni hodnot neznamych koeficientu " }{TEXT 391 1 "A" }{TEXT -1 2 ", " }{TEXT 392 1 "B" }{TEXT -1 2 ", \+ " }{TEXT 393 1 "C" }{TEXT -1 2 ", " }{TEXT 394 2 "D " }{TEXT -1 98 " b udeme potrebovat ctyri rovnice, ktere ziskame postupnym dosazenim cise l 1, 2, 3, 4 za promennou " }{TEXT 399 1 "x" }{TEXT -1 2 ". " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "koef:=solve(\{seq(subs(x=i,d osazeni),i=1..4)\},\{A,B,C,D\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% %koefG<&/%\"AG#!\"\"\"#7/%\"BG\"\"!/%\"DG#\"\"\"\"#O/%\"CGF-" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Dosazenim vypoctenych koeficientu \+ ziskame partikularni reseni." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "part_reseni:=subs(koef,part_reseni);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,part_reseniG*&%\"xG\"\"\",&*&F&F'-%$cosG6#,$F&\"\"$F '#!\"\"\"#7*&#F'\"#OF'-%$sinGF,F'F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Jeho prictenim k obecnemu reseni homogenni rovnice dostan eme obecne reseni nehomogenni rovnice." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "ob_reseni:=y(x)=rhs(ob_res_hom)+simplify(part_reseni) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*ob_reseniG/-%\"yG6#%\"xG,**&&% \"cG6#\"\"\"F/-%$sinG6#,$F)\"\"$F/F/*&&F-6#\"\"#F/-%$cosGF2F/F/*&#F/\" #7F/*&)F)F8F/F9F/F/!\"\"*(#F/\"#OF/F)F/F0F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Vysledek porovname s vystupem procedury " }{TEXT 395 8 "neurkoef" }{TEXT 396 0 "" }{TEXT 397 0 "" }{TEXT -1 28 ", ktera je \+ soucasti souboru " }{TEXT 398 9 "difproc.m" }{TEXT -1 185 ". Tato proc edura hleda obecne reseni nehomogenni rovnice, kterou je mozne resit m etodou neurcitych koeficientu (automatizuje predchozi postup pomoci Ma pleovskeho programovaciho jazyka)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "neurkoef(rovnice2,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,**&&%\"cG6#\"\"\"F--%$sinG6#,$F'\"\"$F-F -*&&F+6#\"\"#F--%$cosGF0F-F-*&#F-\"#7F-*&)F'F6F-F7F-F-!\"\"*(#F-\"#OF- F'F-F.F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Zobrazime grafy nek terych partikularnich reseni pro pevnou hodnotu konstanty " }{XPPEDIT 18 0 "c[1];" "6#&%\"cG6#\"\"\"" }{TEXT -1 5 " = 0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 259 "with(plots): graf1:=plot(subs(c[1]=0,c[2]= -1,rhs(ob_reseni)),x=-4..4,color=black): graf2:=plot(subs(c[1]=0,c[2]= 0,rhs(ob_reseni)),x=-4..4,color=blue): graf3:=plot(subs(c[1]=0,c[2]=1, rhs(ob_reseni)),x=-4..4,color=red): display(graf1,graf2,graf3,thickn ess=2);" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}{PARA 13 "" 1 "" {GLPLOT2D 354 266 266 {PLOTDATA 2 "6(-%'CURVESG6$7[u7$$!\"%\"\"!$!3dup[$y6'G?!#<7$$!3ymmm\"p 0k&RF-$!3e'[\\M/.t$=F-7$$!3cLLL$Q6G\"RF-$!3EE/IveU>;F-7$$!3!******\\2< #pQF-$!3!eC,-T\\$z8F-7$$!3ommmmFiDQF-$!3ug4e0Tr@6F-7$$!3=LL3#f\"p(y$F- $!3-48xkW9q))!#=7$$!37++]=$F-$\"3S=F-7$$!3_LLL[9 cgJF-$\"3>nE)y`pW#=F-7$$!3B++]i/vSJF-$\"3IgIcK2CA=F-7$$!3%pmmmZR47$F-$ \"3zYSG*>!e8=F-7$$!3kLL$3\\G65$F-$\"3i4N\\37f)z\"F-7$$!3M+++0vJ\"3$F-$ \"3?#)=;*=)Rx*)Q'R\"F-7$$!3!******\\`oz$GF-$\" 3k([s^6/o3\"F-7$$!3ummm@\"Rbz#F-$\"3')*)H[*GCT0*FG7$$!3eLLL3(4Jv#F-$\" 3P8w5Lx$p8(FG7$$!3T+++&H!o5FF-$\"3=$=2<&)*o\\^FG7$$!3!omm;)3DoEF-$\"3e ZOy&\\6f7$FG7$$!3$*******Rv&fi#F-$\"3IkOe#F-$!3m ZELV#33'))!#>7$$!32nmmc3PTDF-$!3_8k'o8`u\"GFG7$$!3?+++:v2*\\#F-$!3!>b> s%>QfYFG7$$!3$pmmTrp?T#F-$!3%)=;y6@Ka!)FG7$$!3BLLL8>1DBF-$!35:<3wriv5F -7$$!3%******\\RD%[AF-$!37(H(fh$zVC\"F-7$$!3kmmmw))yr@F-$!3Fadt;SzT8F- 7$$!3ALLeM-A]@F-$!35R4iUO\"F- 7$$!3QmmT]H32@F-$!3Q<(4+djnO\"F-7$$!3SLLL3V^&3#F-$!3!f,@\\@2NO\"F-7$$! 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Pouzijeme tzv. princip superpozice. Vytvorime dv e nove rovnice se stejnymi levymi stranami, pravou stranu \"rozdelime \" tak, aby kazda jeji cast byla typu 1, 2 nebo 3. Dale budeme resit o be ziskane rovnice paralelne. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "rce1:=lhs(rovnice3)=op(1,rhs(rovnice3)); rce2:=lhs(ro vnice3)=op(2,rhs(rovnice3)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rc e1G/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"F*F2-%$sinGF," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rce2G/,&-%%diffG6$-%\"yG6#%\"xG-%\" $G6$F-\"\"#\"\"\"F*F2,$-%$expG6#,$F-!\"#F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Vyjadrime tvary partikularnich reseni. U prvni rovnice je " }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 12 " = 1 a cislo" } {TEXT 405 3 " i " }{TEXT -1 93 "je pritom jednoduchym korenem charakte risticke rovnice. Proto zde nesmime zapomenout na clen " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "part_reseni1:=x^1*(A*cos(x)+B*sin(x)); part_reseni2:=C*exp(-2* x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-part_reseni1G*&%\"xG\"\"\",& *&%\"AGF'-%$cosG6#F&F'F'*&%\"BGF'-%$sinGF-F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-part_reseni2G*&%\"CG\"\"\"-%$expG6#,$%\"xG!\"#F'" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Tvary partikularnich reseni dosad ime do prislusnych rovnic a vypocteme nezname koeficienty. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "dosazeni1:=simplify(subs(y( x)=part_reseni1,rce1)); dosazeni2:=simplify(subs(y(x)=part_reseni2,rce 2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*dosazeni1G/,&*&%\"AG\"\"\"- %$sinG6#%\"xGF)!\"#*(\"\"#F)%\"BGF)-%$cosGF,F)F)F*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%*dosazeni2G/,$*&%\"CG\"\"\"-%$expG6#,$%\"xG!\"#F)\" \"&,$F*F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "koef1:=solve( \{seq(subs(x=i,dosazeni1),i=1..2)\},\{A,B\}); koef2:=solve(dosazeni2, \{C\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&koef1G<$/%\"AG#!\"\"\"\" #/%\"BG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&koef2G<#/%\"CG#!\"# \"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Dosazenim koeficientu do staneme hledana partikularni reseni." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "part_reseni1:=subs(koef1,part_reseni1); part_reseni2: =subs(koef2,part_reseni2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-part _reseni1G,$*&%\"xG\"\"\"-%$cosG6#F'F(#!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-part_reseni2G,$-%$expG6#,$%\"xG!\"##F+\"\"&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 142 "Partikularni reseni zadane rovnic e lze nyni vyjadrit jako soucet obou nalezenych partikularnich reseni. Pak obecne reseni ma nasledujici tvar." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "ob_reseni:=y(x)=rhs(hom(lhs(rovnice3)=0,y(x)))+ (part _reseni1+part_reseni2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*ob_resen iG/-%\"yG6#%\"xG,**&&%\"cG6#\"\"\"F/-%$sinGF(F/F/*&&F-6#\"\"#F/-%$cosG F(F/F/*&#F/F5F/*&F)F/F6F/F/!\"\"*&#F5\"\"&F/-%$expG6#,$F)!\"#F/F;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "Pro nalezeni konkretniho partikul arniho reseni, ktere se pozaduje v zadani, musime dane pocatecni podmi nky dosadit do obecneho reseni a jeho derivace. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "podminka1:=3/5=subs(x=0,rhs(ob_reseni));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*podminka1G/#\"\"$\"\"&,(*&&%\"cG6# \"\"\"F.-%$sinG6#\"\"!F.F.*&&F,6#\"\"#F.-%$cosGF1F.F.*&#F6F(F.-%$expGF 1F.!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "podminka2:=3/10 =subs(x=0,diff(rhs(ob_reseni),x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%*podminka2G/#\"\"$\"#5,**&&%\"cG6#\"\"\"F.-%$cosG6#\"\"!F.F.*&&F,6# \"\"#F.-%$sinGF1F.!\"\"*&#F.F6F.F/F.F9*&#\"\"%\"\"&F.-%$expGF1F.F." }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Ze ziskane soustavy rovnic vypoc teme konstanty " }{XPPEDIT 18 0 "c[1];" "6#&%\"cG6#\"\"\"" }{TEXT -1 3 " a " }{XPPEDIT 18 0 "c[2];" "6#&%\"cG6#\"\"#" }{TEXT -1 39 ". Ty po tom dosadime do obecneho reseni." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "konst:=solve(\{podminka1,podminka2\},\{c[1],c[2]\}); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&konstG<$/&%\"cG6#\"\"\"\"\"!/&F (6#\"\"#F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "part_reseni:= subs(konst,ob_reseni);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,part_rese niG/-%\"yG6#%\"xG,(-%$cosGF(\"\"\"*&#F-\"\"#F-*&F)F-F+F-F-!\"\"*&#F0\" \"&F--%$expG6#,$F)!\"#F-F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Pro nalezeni partikularniho reseni muzeme pouzit take proceduru " }{TEXT 408 8 "neurkoef" }{TEXT -1 16 " nebo proceduru " }{TEXT 409 4 "part" } {TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "podminky: =[[0,3/5],[0,3/10]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)podminkyG7$ 7$\"\"!#\"\"$\"\"&7$F'#F)\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "neurkoef(rovnice3,y(x),podminky);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(-%$cosGF&\"\"\"*&#F+\"\"#F+*&F'F+F)F+F+!\"\"*&#F .\"\"&F+-%$expG6#,$F'!\"#F+F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "part(ob_reseni,podminky);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"yG6#%\"xG,(-%$cosGF&\"\"\"*&#F+\"\"#F+*&F'F+F)F+F+!\"\"*&#F.\"\"&F+- %$expG6#,$F'!\"#F+F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Overime, \+ zda nalezene reseni vyhovuje zadane rovnici a zobrazime ho." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "simplify(subs(part_reseni,ro vnice3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%$sinG6#%\"xG\"\"\"*& \"\"#F)-%$expG6#,$F(!\"#F)!\"\"F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot(rhs(part_reseni),x=-1.3..17.5,thickness=2);" }} {PARA 13 "" 1 "" {GLPLOT2D 354 266 266 {PLOTDATA 2 "6&-%'CURVESG6$7ir7 $$!3/+++++++8!#<$!3>3,xY@7W\\F*7$$!3smmTvLb(>\"F*$!3ggBVJWx/QF*7$$!3UL L$3v1^4\"F*$!33Ml\\(*Q9mGF*7$$!33,+]i7gE**!#=$!3AWrF:$!3ELqA2eqYjF:7$$!3&QLLeg/mL &F:$!30\\'o*[cKNs!#>7$$!3\"3+](y')pJLF:$\"3kB!*H'offB$F:7$$!3xnmm^FzE8 F:$\"3[C4'pE#3a`F:7$$!3Bvm;a\"QCA)FL$\"3IWF=H](4m&F:7$$!3ysmm\"z[p<$FL $\"3YBN%e8<8*eF:7$$\"3qHL$3dS&o=FL$\"3yz(z;F:7$$\"3`mmTO0&zB\"F*$\" 3stJ%35%)e3*FL7$$\"3#****\\F%RxI9F*$\"3!QL[!fB+&o\"FL7$$\"3bmmT&=#>I;F *$!3P()H,=GbKEFL7$$\"3>LL3G/hH=F*$!3)paLv%)H0@$FL7$$\"3Umm\"R8*QG?F*$! 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#;$\"3!)G]M\"[t$fKF*7$$\"3$**\\7T=qU-\"Ffal$\"3^N]>DH#z\"GF*7$$\"3(*** \\()QRKV5Ffal$\"3udZ'*=c9[AF*7$$\"3'**\\i\\))zW1\"Ffal$\"3Cu[S%Q*G&[\" F*7$$\"3#****\\5$ej&3\"Ffal$\"3RmfQe*Q[9'F:7$$\"3;L$e&*QXY5\"Ffal$!3.b N0=tS+BF:7$$\"3Smm1[\\lB6Ffal$!3NoR&p8]@5\"F*7$$\"33LLG4<&R9\"Ffal$!3C *3H]O_r-#F*7$$\"3%*****\\q%[U;\"Ffal$!3Z7:!pDye!HF*7$$\"3[mT&GNSE=\"Ff al$!3cjKZlSNGOF*7$$\"3=L$3_BK5?\"Ffal$!3G-2kD;7^UF*7$$\"3emTb?%R6A\"Ff al$!3S,!Gr#3Q(y%F*7$$\"3)******egY7C\"Ffal$!3Q%p;*HcpW^F*7$$\"3k;/6`iq ]7Ffal$!37\"Hp+Y%HW_F*7$$\"3IL3K+f;g7Ffal$!3E5+4h$HvH&F*7$$\"3aTg#Rs&* [E\"Ffal$!3V#RF***4L1`F*7$$\"3y\\7`Zbip7Ffal$!3L3u7e%zII&F*7$$\"3-ek8r `Nu7Ffal$!3AM>-Uon(G&F*7$$\"3Um;u%>&3z7Ffal$!37oO<%3`+E&F*7$$\"3;L$e(3 8'))H\"Ffal$!3yn\">2)Rv6]F*7$$\"3\"***\\xAuj=8Ffal$!3#R)35`F;_XF*7$$\" 3>L3(*)=szL\"Ffal$!3LjN6#G<$4RF*7$$\"3Ymm;bpId8Ffal$!3'>>O?q*y$4$F*7$$ \"37L3_j'[vP\"Ffal$!3'Q2\\WNcL3#F*7$$\"3y**\\(=P!z(R\"Ffal$!35&z(Qbk#z \\*F:7$$\"3YmT!Ga&G<9Ffal$\"3I%Rqa)Gpr@F:7$$\"3KLLt82yO9Ffal$\"3)f'pA \"RXOT\"F*7$$\"3MLeuF!\\nW\"Ffal$\"3'eYUmQ3>-#F*7$$\"3OL$eR#3[>EF*7$$\"3SL3xbcom9Ffal$\"3u')eE1H1+KF*7$$\"3ULLypRlw9Ffal$ \"3iafp)*RVdPF*7$$\"3)**\\iC^Dk\\\"Ffal$\"3Qz!4t*ogqZF*7$$\"3am;9bq>;: Ffal$\"3*fA)G%[!>CcF*7$$\"3KL3Z\\]OM:Ffal$\"3h<*o[lXRB'F*7$$\"35++!Q/L Db\"Ffal$\"3i1+n`[>]mF*7$$\"3S$eRnoQxb\"Ffal$\"3!o*)=&R8!4t'F*7$$\"3qm \"z'HV%Hc\"Ffal$\"3e&*\\FXZs$z'F*7$$\"3+](=E(*\\\"o:Ffal$\"3bne=\\YNQo F*7$$\"3IL$ebhbLd\"Ffal$\"3urL![MHX'oF*7$$\"3g;z\\e7cy:Ffal$\"3'Hwd9$p .soF*7$$\"3!**\\P9!pw$e\"Ffal$\"3)G3@.F;2'oF*7$$\"3?$3xVas*)e\"Ffal$\" 3VWZV-yXIoF*7$$\"3]mmJ(=yTf\"Ffal$\"33%f^aA/7y'F*7$$\"3%)**\\sV@!Gh\"F fal$\"3)*)p[#)\\-*\\kF*7$$\"3=LL8+hUJ;Ffal$\"3Au4L0sY\")eF*7$$\"3_mT5A JG^;Ffal$\"3I]F*7$$\"3))**\\2W,9r;Ffal$\"3_Mp)ym(*H&RF*7$$\"3') *\\Pf*Hk!o\"Ffal$\"3kh%[A6#=oLF*7$$\"3')****zZe9!p\"Ffal$\"3(fLaPvV\\u #F*7$$\"3')*\\i'*p['*p\"Ffal$\"3W+%o$RMb)3#F*7$$\"3&)**\\_^::4 " 0 "" {MPLTEXT 1 0 8 " restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "rovnice4:=diff (y(x),x$5)-2*diff(y(x),x$4)-2*diff(y(x),x$3)+4*diff(y(x),x$2)+diff(y(x ),x)-2*y(x)=25*exp(-x)*cos(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)r ovnice4G/,.-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"&\"\"\"*&\"\"#F2-F(6$F *-F/6$F-\"\"%F2!\"\"*&F4F2-F(6$F*-F/6$F-\"\"$F2F:*&F9F2-F(6$F*-F/6$F-F 4F2F2-F(6$F*F-F2*&F4F2F*F2F:,$*&-%$expG6#,$F-F:F2-%$cosGF,F2\"#D" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "podminky:=[[0,1],[0,-3],[0,5 /2],[0,-4],[0,-7/2]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)podminkyG7 '7$\"\"!\"\"\"7$F'!\"$7$F'#\"\"&\"\"#7$F'!\"%7$F'#!\"(F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 151 "Jde o rovnici 5. radu, kterou lze resit \+ metodou neurcitych koeficientu. Koreny charakteristicke rovnice lze zi skat vytykanim a pouzitim znamych vzorcu." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 52 "read \"difproc.m\": solve(char(lhs(rovnice4)=0,y(x) ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"#!\"\"F$\"\"\"F%" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Pomoci procedury " }{TEXT 423 9 "n eurkoef " }{TEXT -1 39 "dostaneme obecne reseni zadane rovnice." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "ob_reseni:=neurkoef(rovnice4 ,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*ob_reseniG/-%\"yG6#%\"xG ,0*&&%\"cG6#\"\"\"F/-%$expG6#,$F)!\"\"F/F/*(&F-6#\"\"#F/F)F/F0F/F/*&&F -6#\"\"$F/-F1F(F/F/*(&F-6#\"\"%F/F)F/F=F/F/*&&F-6#\"\"&F/-F16#,$F)F8F/ F/*(#F/F8F/F0F/-%$cosGF(F/F/*&#F " 0 "" {MPLTEXT 1 0 38 "part_reseni:=part(ob_reseni,podminky);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%,part_reseniG/-%\"yG6#%\"xG,*-%$expG 6#,$F)!\"\"#\"\"\"\"\"#*&#F1F2F1*&F)F1-F,F(F1F1F/*(F0F1F+F1-%$cosGF(F1 F1*&#\"\"$F2F1*&F+F1-%$sinGF(F1F1F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Overime, zda nalezene reseni vyhovuje dane rovnici a zobrazime \+ jej pro vhodny rozsah promenne " }{TEXT 424 1 "x" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "simplify(subs(part_reseni,ro vnice4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&-%$expG6#,$%\"xG!\" \"\"\"\"-%$cosG6#F*F,\"#DF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot(rhs(part_reseni),x=-3.4..2.3,thickness=2);" }}{PARA 13 "" 1 " " {GLPLOT2D 354 266 266 {PLOTDATA 2 "6&-%'CURVESG6$7in7$$!3!********** ****R$!#<$!3I\"oWAIUJ4\"!#;7$$!3g*\\7`0R*oLF*$!3+gR.N;+*Q*F*7$$!3s**\\ i5\"yyL$F*$!3'>&yBE283zF*7$$!3')*\\Pf;%)G^F*7$$!3)*\\(ouH/;a\"HJF*7$$!3'*****\\8$))[)HF*$\"3y/@)*Q<1G[F*7$$!3u**\\7O$) pBHF*$\"3zzXku)oIL'F*7$$!3m\\7.%=*ziGF*$\"39bsMFp,[wF*7$$!3/+v$>.+>!GF *$\"3I^WEZ(p.z)F*7$$!33+DJA(Qau#F*$\"351/\\!QXSq*F*7$$!37+vo7u(*)o#F*$ \"37!))RJo2'[5F-7$$!39+v=\"e_?d#F*$\"3;y&))*Q,=s6F-7$$!3!**\\(=n1f6DF* $\"3\"Rs3(f.j<7F-7$$!37+v=`(G6X#F*$\"3Q3@\")>Jw^7F-7$$!3E+DJT2'3R#F*$ \"3+V7!>g]aF\"F-7$$!3'**\\P%HFfIBF*$\"3[lFnwjr*G\"F-7$$!3'*\\PMr'*foAF *$\"3M_'=47saH\"F-7$$!3'****\\Kh1m?#F*$\"3#zF*$\"3%y(QK&zefA\"F-7$$!3E++DYA,^=F*$\"3SfWkH%4D;\"F-7$$!37++D&3 \\?t\"F*$\"3gq)e!=IR*3\"F-7$$!3@+v=Q\">Si\"F*$\"3.7y1&z?h,\"F-7$$!3G++ D#3hb\\\"F*$\"3A](y,2E+C*F*7$$!3?+++f-u'Q\"F*$\"3p-/S:_kT%)F*7$$!3E+vo rd:g7F*$\"3*ySsOu'Q;vF*7$$!3E++]^q5[6F*$\"3YDqJ_)\\pr'F*7$$!3R+voY=/AveF*7$$!37+]i+Y3\"3*FL$\"3RKs2v#)*y6&F*7$$!3Y++vo4lfyFL$ \"3!)HC1M2u!Q%F*7$$!30,]P/O*zt'FL$\"3o'R_:Er]v$F*7$$!3j.+D1Y8GbFL$\"3 \"\\x.\\[9n8$F*7$$!37.](=8J9F%FL$\"3OjXoht(eb#F*7$$!3q0]7jqYxJFL$\"3'R ,6)[%)=*4#F*7$$!3m.+v)*e&f*>FL$\"3'Q\"f+\"))ROl\"F*7$$!3!*[++]DR`x!#>$ \"3hJD$G]@/C\"F*7$$\"3)3#**\\i*fz=%F_x$\"3C.Y8a;/l()FL7$$\"3U%*\\7`K=u :FL$\"3gfqM*\\U.c&FL7$$\"3S&**\\P\"Q/dGFL$\"3yInbw%*p%G#FL7$$\"3n!**** \\3^(4SFL$!3z)yS23!)[$[F_x7$$\"3A&***\\P@^S_FL$!3]HU=m\\tXLFL7$$\"3S*) \\7yXwbjFL$!3@53K1E:PfFL7$$\"3s****\\s+.vvFL$!3IG([NREL'))FL7$$\"3M%* \\7$)QEA()FL$!39v7Z7/,!=\"F*7$$\"3k(*\\ivPX@**FL$!3P\"p`Noin^\"F*7$$\" 3#))**\\(3\")Q46F*$!3Uj,_%)*Qn)=F*7$$\"3@*\\7GQI@B\"F*$!3rE/AbVcIBF*7$ $\"3C*****4yX.N\"F*$!3cLbl5)Qn#GF*7$$\"3)*)*\\()*4Q7Z\"F*$!3wf+Oam')>M F*7$$\"3q*\\7$p$H6f\"F*$!3;'yVs\"4O6TF*7$$\"3Q****\\%*pH,C1! RW&[F*7$$\"3G**\\PD:cF=F*$!3G(z*>f\"[e&eF*7$$\"3H*****p-%\\S>F*$!3:q*z U))p+\"pF*7$$\"33*\\7$GO!41#F*$!3[*p#pzr-F#)F*7$$\"3Y*\\(o!)4:w@F*$!3t #)RZ%=*)Rq*F*7$$\"3#)*************H#F*$!3[v=kuDdc6F--%'COLOURG6&%$RGBG $\"#5!\"\"$\"\"!F^_lF]_l-%+AXESLABELSG6$Q\"x6\"Q!6\"-%*THICKNESSG6#\" \"#-%%VIEWG6$;$!#MF\\_l$\"#BF\\_l%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Ukazky aplikaci rovnic 2. radu" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "P\370\355klad 12 - line\341rn\355 oscil \341tor" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 474 0 "" }} {PARA 0 "" 0 "" {TEXT -1 109 "V tomto prikladu se budeme se podrobneji zabyvat rovnici linearniho oscilatoru, coz je rovnice 2. radu tvaru \+ " }{TEXT 426 3 "y''" }{TEXT -1 4 " + 2" }{TEXT 427 3 "ay'" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "b^2;" "6#*$%\"bG\"\"#" }{TEXT -1 0 "" }{TEXT 428 1 "y" }{TEXT -1 10 " = 0, kde " }{XPPEDIT 18 0 "0 <= a;" "6#1\"\"! %\"aG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res tart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "rovnice:=diff(y(x) ,x$2)+2*a*diff(y(x),x)+b^2*y(x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%(rovniceG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*(F1F2%\"aGF 2-F(6$F*F-F2F2*&)%\"bGF1F2F*F2F2\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Urcime charakteristickou rovnici a vypocteme jeji koreny. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "read \"difproc.m\": \+ char_rovnice:=char(rovnice,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-char_rovniceG/,(*$)%'lambdaG\"\"#\"\"\"F+*(F*F+%\"aGF+F)F+F+* $)%\"bGF*F+F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solve (char_rovnice,lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&%\"aG!\"\" *$-%%sqrtG6#,&*$)F$\"\"#\"\"\"F.*$)%\"bGF-F.F%F.F.,&F$F%F&F%" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Nyni provedeme diskuzi reseni v za vislosti na parametrech " }{TEXT 429 1 "a" }{TEXT -1 2 ", " }{TEXT 430 1 "b" }{TEXT -1 1 "." }{TEXT 431 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 432 3 "I. " }{TEXT -1 9 "Jestl ize " }{TEXT 433 3 "a=0" }{TEXT -1 2 ", " }{TEXT 434 1 " " }{TEXT -1 21 "je dana rovnice tvaru" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "rovnice1:=subs(a=0,rovnice);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% )rovnice1G/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&)%\"bGF1F2F *F2F2\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Urcime koreny chara kteristicke rovnice a obecne reseni." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "solve(char(rovnice1,y(x)),lambda);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$*&^#\"\"\"F%%\"bGF%*&^#!\"\"F%F&F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "ob_reseni1:=y(x)=c[1]*cos(b*x)+c[2]*sin(b *x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+ob_reseni1G/-%\"yG6#%\"xG,& *&&%\"cG6#\"\"\"F/-%$cosG6#*&%\"bGF/F)F/F/F/*&&F-6#\"\"#F/-%$sinGF2F/F /" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Vzorec obecneho reseni lze u pravit na vhodnejsi tvar, kde " }{TEXT 435 1 "A" }{TEXT -1 22 " > 0. J edna se o tzv. " }{TEXT 438 16 "harmonicke kmity" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "harmonicke:=y(x)=A*sin(b*x+p hi);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%+harmonickeG/-%\"yG6#%\"xG*&%\"AG\"\"\"-%$sinG6#,&*&%\"bGF,F)F,F,% $phiGF,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Budeme uvazovat tyto zakladni hodnoty parametru: " }{TEXT 436 2 "A " }{TEXT -1 5 "= 1, " } {TEXT 437 1 "b" }{TEXT -1 6 " = 1, " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 117 " = 0. Budeme postupne animovat reseni tak, ze jeden z p arametru budeme menit ve vhodnem rozsahu. Pri zmene parametru " } {TEXT 439 2 "A " }{TEXT -1 23 "bude dochazet ke zmene " }{TEXT 440 16 "amlitudy kmitani" }{TEXT -1 22 ", pri zmene parametru " }{TEXT 441 1 "b" }{TEXT -1 16 " dojde ke zmene " }{TEXT 442 18 "frekvence kmitani \+ " }{TEXT -1 7 "(resp. " }{TEXT 443 7 "periody" }{TEXT -1 24 ") a pri z mene parametru " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 16 " dojde ke zmene " }{TEXT 444 12 "faze kmitani" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "with(plots): animate(subs(b=1,phi= 0,rhs(harmonicke)),x=-1.5..11,A=.5..4,scaling=constrained,view=[-1.5.. 11,-4..4],thickness=2,tickmarks=[6,4]); #amplituda" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }} {PARA 13 "" 1 "" {GLPLOT2D 354 266 266 {PLOTDATA 2 "6(-%(ANIMATEG627#- %'CURVESG6$7S7$$!#:!\"\"$!3As--L\\Z()\\!#=7$$!+LLeR7!\"*$!3YQ[%y#HCGZF 17$$!+nmm\"z*!#5$!3ibYh$[k,:%F17$$!+++](=(F;$!3yOGb**>A#H$F17$$!*LLLe% F5$!3]#Hm6NqA@#F17$$!*nm\"z>F5$!3v!o$3)ya8$)*!#>7$$\")++]iF5$\"3\"R->@ *e'H7$FM7$$\"*nm\"HKF5$\"3MV9(*4\"pme\"F17$$\"*LLL$eF5$\"3o][Vza/aFF17 $$\"*++vV)F5$\"3nszn>VqNPF17$$\"+nm;/6F5$\"3c55M6![aY%F17$$\"+LLek8F5$ \"3!H$)pLpmS*[F17$$\"++++D;F5$\"31e\"*p-nl#*\\F17$$\"+nmT&)=F5$\"3aD$Q 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