{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "P\370\355klad 10" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Naleznete partikularni reseni rovnice " }{TEXT 257 6 "y'' + " }{TEXT -1 1 "9" }{TEXT 260 4 "y = " }{TEXT -1 4 "sin(" }{TEXT 258 1 "x" } {TEXT -1 2 ") " }{TEXT 259 2 "- " }{TEXT -1 1 "2" }{XPPEDIT 18 0 "exp( -2*x);" "6#-%$expG6#,$*&\"\"#\"\"\"%\"xGF)!\"\"" }{TEXT -1 30 " splnuj ici pocatecni podminky " }{TEXT 262 1 "y" }{TEXT -1 11 "(0) = 3/5, " } {TEXT 263 2 "y'" }{TEXT -1 11 "(0) = 3/10." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "rovnice3:=diff(y(x),x$2)+y(x)=sin(x)-2*exp(-2*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)rovnice3G/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\" \"#\"\"\"F*F2,&-%$sinGF,F2*&F1F2-%$expG6#,$F-!\"#F2!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Vypocteme koreny charakteristicke rovnice ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "read \"difproc.m\": so lve(char(lhs(rovnice3)=0,y(x)),lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$^#\"\"\"^#!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 295 "Prava s trana rovnice je typu 4, presneji je souctem pravych stran typu 3 a 2. Pouzijeme tzv. princip superpozice. Vytvorime dve nove rovnice se ste jnymi levymi stranami, pravou stranu \"rozdelime\" tak, aby kazda jeji cast byla typu 1, 2 nebo 3. Dale budeme resit obe ziskane rovnice par alelne. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "rce1:=lhs(rov nice3)=op(1,rhs(rovnice3)); rce2:=lhs(rovnice3)=op(2,rhs(rovnice3)); \+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rce1G/,&-%%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 par tikularnich reseni. U prvni rovnice je " }{XPPEDIT 18 0 "beta;" "6#%%b etaG" }{TEXT -1 12 " = 1 a cislo" }{TEXT 261 3 " i " }{TEXT -1 93 "je \+ pritom jednoduchym korenem charakteristicke 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 dosadime do prislusnych rovnic a vypoc teme nezname koeficienty. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "dosazeni1:=simplify(subs(y(x)=part_reseni1,rce1)); dosazeni2:=sim plify(subs(y(x)=part_reseni2,rce2));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%*dosazeni1G/,&*&%\"AG\"\"\"-%$sinG6#%\"xGF)!\"#*(\"\"#F)%\"BGF)-%$ cosGF,F)F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*dosazeni2G/,$*&%\"C G\"\"\"-%$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<$/%\"AG#!\"\"\"\"#/%\"BG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&koef2G<#/%\"CG#!\"#\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Dosazenim koeficientu dostaneme hledana partikularni rese ni." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "part_reseni1:=subs(k oef1,part_reseni1); part_reseni2:=subs(koef2,part_reseni2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-part_reseni1G,$*&%\"xG\"\"\"-%$cosG6#F'F( #!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-part_reseni2G,$-%$ex pG6#,$%\"xG!\"##F+\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 142 "Parti kularni reseni zadane rovnice lze nyni vyjadrit jako soucet obou nalez enych partikularnich reseni. Pak obecne reseni ma nasledujici tvar." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "ob_reseni:=y(x)=rhs(hom(lh s(rovnice3)=0,y(x)))+ (part_reseni1+part_reseni2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%*ob_reseniG/-%\"yG6#%\"xG,**&&%\"cG6#\"\"\"F/-%$sin GF(F/F/*&&F-6#\"\"#F/-%$cosGF(F/F/*&#F/F5F/*&F)F/F6F/F/!\"\"*&#F5\"\"& F/-%$expG6#,$F)!\"#F/F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "Pro n alezeni konkretniho partikularniho reseni, ktere se pozaduje v zadani, musime dane pocatecni podminky dosadit do obecneho reseni a jeho deri vace. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "podminka1:=3/5= subs(x=0,rhs(ob_reseni));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*podmin ka1G/#\"\"$\"\"&,(*&&%\"cG6#\"\"\"F.-%$sinG6#\"\"!F.F.*&&F,6#\"\"#F.-% $cosGF1F.F.*&#F6F(F.-%$expGF1F.!\"\"" }}}{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 vypocteme konstanty " }{XPPEDIT 18 0 "c[1]; " "6#&%\"cG6#\"\"\"" }{TEXT -1 3 " a " }{XPPEDIT 18 0 "c[2];" "6#&%\"c G6#\"\"#" }{TEXT -1 39 ". Ty potom dosadime do obecneho reseni." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "konst:=solve(\{podminka1,pod minka2\},\{c[1],c[2]\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&konstG< $/&%\"cG6#\"\"\"\"\"!/&F(6#\"\"#F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "part_reseni:=subs(konst,ob_reseni);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,part_reseniG/-%\"yG6#%\"xG,(-%$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 264 8 "neurkoef" }{TEXT -1 16 " nebo p roceduru " }{TEXT 265 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),podmi nky);" }}{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 zad ane rovnici a zobrazime ho." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "simplify(subs(part_reseni,rovnice3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%$sinG6#%\"xG\"\"\"*&\"\"#F)-%$expG6#,$F(!\"#F)!\" \"F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot(rhs(part_resen i),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$$!3ULL$3v1^4\"F*$!33Ml\\(*Q9mGF*7$$!33,+ ]i7gE**!#=$!3AWrF:$!3ELqA2eqYjF:7$$!3&QLLeg/mL&F:$!30\\'o*[cKNs!#>7$$!3\"3+](y')p JLF:$\"3kB!*H'offB$F:7$$!3xnmm^FzE8F:$\"3[C4'pE#3a`F:7$$!3Bvm;a\"QCA)F L$\"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$$\"3Umm\"R8*QG?F*$!3T#GMs\\l&3l!#@7$$\"3))***\\(Ry;FAF *$\"35Q\"['>bHmkFL7$$\"3YL$3<-!>\\F:7$$\"3)om;aPT!*> $F*$\"3ah!*[culyfF:7$$\"3N+++6my,MF*$\"3/0qDSQdonF:7$$\"3H++]uad.NF*$ \"3e&4)ov42FqF:7$$\"3A+++QVO0OF*$\"3+WJj_w+wrF:7$$\"3;++],K:2PF*$\"3;% H))f#Q&F:7$$\"3%omT&QkYzVF*$\"3s-X0`r \\()QF:7$$\"3mLL3)3@wb%F*$\"3W)z4WRc3(>F:7$$\"3_mmm2tI\")\\F*$!3gx%*Gl #)ogRF:7$$\"3;nmm5^AS`F*$!3(>ZqU;[-\")*F:7$$\"3+++vS=tddF*$!39n,#>X/ai \"F*7$$\"3]K$32%R^UfF*$!3%e5K`psz&=F*7$$\"3*emm1/'HFhF*$!3Y'zd1$eiQ?F* 7$$\"3&)*\\(olOmGiF*$!3CSkaD;>6@F*7$$\"3\"QL32HJ+L'F*$!3!G#3bO#F*7$$ \"3K%eR#GF3#['F*$!33M%H&R-'o>#F*7$$\"3&3+]2amF`'F*$!3#\\^'pkB;'>#F*7$$ \"3j+]7\"\\#HHmF*$!3/R!zkT*Qx@F*7$$\"3T++]T%=es'F*$!3&*y,F!3*=N@F*7$$ \"3>+](=RWB#oF*$!3s'e:d_D\"p?F*7$$\"3(****\\AMq)=pF*$!3'*=\\>(=A!z>F*7 $$\"3nL$3#[(*H?rF*$!3&[;&3L!*H9\"*yF*$\"3w'H8W&y/'4\" F:7$$\"3iKL$e^?24)F*$\"37:%f.L'RUrF:7$$\"3KL$e>NmzH)F*$\"3m^x8e\"=EN\" F*7$$\"3CKL3)=7_])F*$\"3Ab6Q:yhr>F*7$$\"3ammm(H?co)F*$\"33+(\\l%zXqCF* 7$$\"33***\\sSGg'))F*$\"3jW(QS#o\"4\"HF*7$$\"3Qm;/3T(31*F*$\"3Y)=!4.XC *H$F*7$$\"3oLL$)3)>dD*F*$\"3uG.fY*Rhd$F*7$$\"3'))*\\7YrOc$*F*$\"3CVI+/ *z&pOF*7$$\"3%em;M[9qX*F*$\"3ZFl)p3qls$F*7$$\"3W)\\i?:Qt]*F*$\"37ZR$p[ $)3u$F*7$$\"3#GL32#=md&*F*$\"33\"Qa*[k^XPF*7$$\"3TlTN*[&)zg*F*$\"3a4\" o\\gI.u$F*7$$\"3+)****z:4$e'*F*$\"3tm2p'H8_s$F*7$$\"3U)**\\dsO_&)*F*$ \"3+')3m_!>$pNF*7$$\"3*)***\\$Hk@05!#;$\"3!)G]M\"[t$fKF*7$$\"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$$\"3#****\\5$ej&3\"Ffal$\"3Rmf Qe*Q[9'F:7$$\"3;L$e&*QXY5\"Ffal$!3.bN0=tS+BF:7$$\"3Smm1[\\lB6Ffal$!3No R&p8]@5\"F*7$$\"33LLG4<&R9\"Ffal$!3C*3H]O_r-#F*7$$\"3%*****\\q%[U;\"Ff al$!3Z7:!pDye!HF*7$$\"3[mT&GNSE=\"Ffal$!3cjKZlSNGOF*7$$\"3=L$3_BK5?\"F fal$!3G-2kD;7^UF*7$$\"3emTb?%R6A\"Ffal$!3S,!Gr#3Q(y%F*7$$\"3)******egY 7C\"Ffal$!3Q%p;*HcpW^F*7$$\"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$e(38'))H\"Ffal$!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%Rq a)Gpr@F:7$$\"3KLLt82yO9Ffal$\"3)f'pA\"RXOT\"F*7$$\"3MLeuF!\\nW\"Ffal$ \"3'eYUmQ3>-#F*7$$\"3OL$eR#3[>EF*7$$\"3SL3xbcom9Ffa l$\"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$$\"35++!Q/LDb\"Ffal$\"3i1+n`[>]mF*7$$\"3S$eR noQxb\"Ffal$\"3!o*)=&R8!4t'F*7$$\"3qm\"z'HV%Hc\"Ffal$\"3e&*\\FXZs$z'F* 7$$\"3+](=E(*\\\"o:Ffal$\"3bne=\\YNQoF*7$$\"3IL$ebhbLd\"Ffal$\"3urL![M HX'oF*7$$\"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\"Ffal$\"3)*)p[#)\\-*\\kF*7$$\"3=LL8 +hUJ;Ffal$\"3Au4L0sY\")eF*7$$\"3_mT5AJG^;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 0 "" }}}}{MARK "0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }