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{SECT 0 {EXCHG {PARA 3 "" 0 "" {TEXT 207 35 "Solving and graphing ODE
, part II\n\n" }{TEXT 200 179 "Maple worksheet (Worksheet #2 for Math \+
3260). More info about solving linear ODEs, and graphing their solutio
ns.\nCreated by A. Foster in January 1998, last modified February 2012
." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 189 "\nSometimes, when it is di
fficult or impossible to solve a differential equation explicitly, an \+
implicit solution is given instead. Consider the nonlinear equation dy
/dx = (y - x)/(2x - y). " }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "`
:=`(ode1, diff(y(x), x) = `/`(`*`(`+`(y(x), `-`(x))), `*`(`+`(`*`(2, `
*`(x)), `-`(y(x))))));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_s
yslibGF'6)-I#miGF$6%Q%ode1F'/%'italicGQ%trueF'/%,mathvariantGQ'italicF
'-I#moGF$6-Q*≔F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=
/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accent
GF=/%'lspaceGQ,0.2777778emF'/%'rspaceGFL-F,6%Q%diffF'F/F2-I(mfencedGF$
6$-F#6&-F,6%Q\"yF'F/F2-FS6$-F#6#-F,6%Q\"xF'F/F2F9-F66-Q\",F'F9F;/F?F1F
@FBFDFFFH/FKQ&0.0emF'/FNQ,0.3333333emF'FhnF9-F66-Q\"=F'F9F;F>F@FBFDFFF
HFJFM-I&mfracGF$6(-FS6$-F#6&FWFZ-F66-Q(−F'F9F;F>F@FBFDFFFH/FKQ,0
.2222222emF'/FNFapFhnF9-F#6)-I#mnGF$6$Q\"2F'F9-F66-Q'⋅F'F9F;F>F@F
BFDFFFHF_o/FNF`oFhnF]pFWFZ-F,6#Q!F'/%.linethicknessGQ\"1F'/%+denomalig
nGQ'centerF'/%)numalignGFeq/%)bevelledGF=-F66-Q\";F'F9F;F^oF@FBFDFFFHF
_oFM" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "`:=`(soln, dsolve(ode
1));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I#miGF$
6%Q%solnF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q*&colo
neq;F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*s
ymmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%'lspaceGQ,0.2
777778emF'/%'rspaceGFL-F,6%Q'dsolveF'F/F2-I(mfencedGF$6$-F#6#-F,6%Q%od
e1F'F/F2F9-F66-Q\";F'F9F;/F?F1F@FBFDFFFH/FKQ&0.0emF'FM" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 208 61 "Maple is unable to solve explicitly for \+
y as a function of x." }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "`:=`
(soln1, dsolve(ode1, y(x), explicit));" "-I%mrowG6#/I+modulenameG6\"I,
TypesettingGI(_syslibGF'6'-I#miGF$6%Q&soln1F'/%'italicGQ%trueF'/%,math
variantGQ'italicF'-I#moGF$6-Q*≔F'/F3Q'normalF'/%&fenceGQ&false
F'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablel
imitsGF=/%'accentGF=/%'lspaceGQ,0.2777778emF'/%'rspaceGFL-F,6%Q'dsolve
F'F/F2-I(mfencedGF$6$-F#6(-F,6%Q%ode1F'F/F2-F66-Q\",F'F9F;/F?F1F@FBFDF
FFH/FKQ&0.0emF'/FNQ,0.3333333emF'-F,6%Q\"yF'F/F2-FS6$-F#6#-F,6%Q\"xF'F
/F2F9FZ-F,6%Q)explicitF'F/F2F9-F66-Q\";F'F9F;FgnF@FBFDFFFHFhnFM" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 208 182 "In this situation, there are som
e general things we can try with Maple. We can plot the direction fiel
d. Also, we can get some indication of the true solution by plotting n
umerical  " }}{PARA 0 "" 0 "" {TEXT 208 151 "approximations for partic
ular solutions using the \"DEplot\" command in the Maple package \"DEt
ools\". Further analysis is possible for specific equations." }}}
{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "with(DEtools):" "-I%mrowG6#/I+
modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q%withF'/%'italicG
Q%trueF'/%,mathvariantGQ'italicF'-I(mfencedGF$6$-F#6#-F,6%Q(DEtoolsF'F
/F2/F3Q'normalF'-I#moGF$6-Q\":F'F=/%&fenceGQ&falseF'/%*separatorGFE/%)
stretchyGFE/%*symmetricGFE/%(largeopGFE/%.movablelimitsGFE/%'accentGFE
/%'lspaceGQ,0.2777778emF'/%'rspaceGFT" }}}{EXCHG {PARA 0 "> " 0 "" 
{XPPEDIT 19 1 "dfieldplot(ode1, y(x), x = -3 .. 3, y = -3 .. 3);" "-I%
mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q+dfield
plotF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I(mfencedGF$6$-F#64-
F,6%Q%ode1F'F/F2-I#moGF$6-Q\",F'/F3Q'normalF'/%&fenceGQ&falseF'/%*sepa
ratorGF1/%)stretchyGFE/%*symmetricGFE/%(largeopGFE/%.movablelimitsGFE/
%'accentGFE/%'lspaceGQ&0.0emF'/%'rspaceGQ,0.3333333emF'-F,6%Q\"yF'F/F2
-F66$-F#6#-F,6%Q\"xF'F/F2FAF=Fin-F>6-Q\"=F'FAFC/FGFEFHFJFLFNFP/FSQ,0.2
777778emF'/FVFao-F>6-Q*&uminus0;F'FAFCF_oFHFJFLFNFP/FSQ,0.2222222emF'/
FVFgo-I#mnGF$6$Q\"3F'FA-F>6-Q#..F'FAFCF_oFHFJFLFNFPFfo/FVFTFioF=FXF\\o
FcoFioF]pFioFA-F>6-Q\";F'FAFCFFFHFJFLFNFPFRFbo" }}}{EXCHG {PARA 0 "> "
 0 "" {XPPEDIT 19 1 "DEplot(ode1, y(x), x = -2 .. 4, [[y(0) = 2]], lin
ecolor = black);" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibG
F'6%-I#miGF$6%Q'DEplotF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I(
mfencedGF$6$-F#63-F,6%Q%ode1F'F/F2-I#moGF$6-Q\",F'/F3Q'normalF'/%&fenc
eGQ&falseF'/%*separatorGF1/%)stretchyGFE/%*symmetricGFE/%(largeopGFE/%
.movablelimitsGFE/%'accentGFE/%'lspaceGQ&0.0emF'/%'rspaceGQ,0.3333333e
mF'-F,6%Q\"yF'F/F2-F66$-F#6#-F,6%Q\"xF'F/F2FAF=Fin-F>6-Q\"=F'FAFC/FGFE
FHFJFLFNFP/FSQ,0.2777778emF'/FVFao-F>6-Q*&uminus0;F'FAFCF_oFHFJFLFNFP/
FSQ,0.2222222emF'/FVFgo-I#mnGF$6$Q\"2F'FA-F>6-Q#..F'FAFCF_oFHFJFLFNFPF
fo/FVFT-Fjo6$Q\"4F'FAF=-F66&-F#6$-F66&-F#6&FX-F66$-F#6#-Fjo6$Q\"0F'FAF
AF\\oFioFA/%%openGQ\"[F'/%&closeGQ\"]F'-F,6#Q!F'FAFcqFfqF=-F,6%Q*linec
olorF'F/F2F\\o-F,6%Q&blackF'F/F2FA-F>6-Q\";F'FAFCFFFHFJFLFNFPFRFbo" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT 208 236 "Hey, Maple had a problem with t
hat numerical \"solution\"! Can you see why, by looking at the equatio
n and the direction field? It cannot follow the direction field near t
he middle of the graph, where the slopes are changing very rapidly." }
}{PARA 0 "" 0 "" {TEXT 208 0 "" }}{PARA 0 "" 0 "" {TEXT 208 170 "Now l
et's look at examples of second order linear equations.  First conside
r y'' + y' - 12y = 20 e^x. In Maple, the second derivative can be writ
ten as \"diff(y(x), x$2)\"." }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1
 "`:=`(ode2, `+`(diff(y(x), `$`(x, 2)), diff(y(x), x), `-`(`*`(12, `*`
(y(x))))) = `+`(`*`(20, `*`(exp(x)))));" "-I%mrowG6#/I+modulenameG6\"I
,TypesettingGI(_syslibGF'64-I#miGF$6%Q%ode2F'/%'italicGQ%trueF'/%,math
variantGQ'italicF'-I#moGF$6-Q*≔F'/F3Q'normalF'/%&fenceGQ&false
F'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablel
imitsGF=/%'accentGF=/%'lspaceGQ,0.2777778emF'/%'rspaceGFL-F,6%Q%diffF'
F/F2-I(mfencedGF$6$-F#6(-F,6%Q\"yF'F/F2-FS6$-F#6#-F,6%Q\"xF'F/F2F9-F66
-Q\",F'F9F;/F?F1F@FBFDFFFH/FKQ&0.0emF'/FNQ,0.3333333emF'Fhn-F66-Q\"$F'
F9F;F>F@FBFDFFFHF_o/FNF`o-I#mnGF$6$Q\"2F'F9F9-F66-Q\"+F'F9F;F>F@FBFDFF
FH/FKQ,0.2222222emF'/FNF_pFO-FS6$-F#6&FWFZF[oFhnF9-F66-Q(−F'F9F;
F>F@FBFDFFFHF^pF`p-Fho6$Q#12F'F9-F66-Q'⋅F'F9F;F>F@FBFDFFFHF_oFfoF
WFZ-F66-Q\"=F'F9F;F>F@FBFDFFFHFJFM-Fho6$Q#20F'F9F[q-F,6%Q$expF'/F0F=F9
FZ-F66-Q\";F'F9F;F^oF@FBFDFFFHF_oFM" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
208 24 "The general solution is:" }}}{EXCHG {PARA 0 "> " 0 "" 
{XPPEDIT 19 1 "dsolve(ode2, y(x));" "-I%mrowG6#/I+modulenameG6\"I,Type
settingGI(_syslibGF'6%-I#miGF$6%Q'dsolveF'/%'italicGQ%trueF'/%,mathvar
iantGQ'italicF'-I(mfencedGF$6$-F#6&-F,6%Q%ode2F'F/F2-I#moGF$6-Q\",F'/F
3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF1/%)stretchyGFE/%*symmetric
GFE/%(largeopGFE/%.movablelimitsGFE/%'accentGFE/%'lspaceGQ&0.0emF'/%'r
spaceGQ,0.3333333emF'-F,6%Q\"yF'F/F2-F66$-F#6#-F,6%Q\"xF'F/F2FAFA-F>6-
Q\";F'FAFCFFFHFJFLFNFPFR/FVQ,0.2777778emF'" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 208 128 "To find a particular solution for this second order equ
ation, we require two initial conditions, such as y(0)=2 and y'(0)= -2
0. " }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "dsolve(\{ode2, y(0) = \+
2, (D(y))(0) = -20\}, y(x));" "-I%mrowG6#/I+modulenameG6\"I,Typesettin
gGI(_syslibGF'6%-I#miGF$6%Q'dsolveF'/%'italicGQ%trueF'/%,mathvariantGQ
'italicF'-I(mfencedGF$6$-F#6&-F66&-F#6/-F,6%Q%ode2F'F/F2-I#moGF$6-Q\",
F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF1/%)stretchyGFI/%*symme
tricGFI/%(largeopGFI/%.movablelimitsGFI/%'accentGFI/%'lspaceGQ&0.0emF'
/%'rspaceGQ,0.3333333emF'-F,6%Q\"yF'F/F2-F66$-F#6#-I#mnGF$6$Q\"0F'FEFE
-FB6-Q\"=F'FEFG/FKFIFLFNFPFRFT/FWQ,0.2777778emF'/FZFfo-F^o6$Q\"2F'FEFA
-F,6%Q\"DF'/F0FIFE-F66$-F#6#FfnFEFinFao-FB6-Q*&uminus0;F'FEFGFdoFLFNFP
FRFT/FWQ,0.2222222emF'/FZFgp-F^o6$Q#20F'FEFE/%%openGQ\"|frF'/%&closeGQ
\"|hrF'FAFfn-F66$-F#6#-F,6%Q\"xF'F/F2FEFE-FB6-Q\";F'FEFGFJFLFNFPFRFTFV
Fgo" }}{PARA 0 "" 0 "" {TEXT 200 36 "We can plot any particular soluti
on." }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "DEplot(ode2, y(x), x =
 -1 .. 2, [[y(0) = 2, (D(y))(0) = -20]], linecolor = blue);" "-I%mrowG
6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q'DEplotF'/%'
italicGQ%trueF'/%,mathvariantGQ'italicF'-I(mfencedGF$6$-F#63-F,6%Q%ode
2F'F/F2-I#moGF$6-Q\",F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF1/
%)stretchyGFE/%*symmetricGFE/%(largeopGFE/%.movablelimitsGFE/%'accentG
FE/%'lspaceGQ&0.0emF'/%'rspaceGQ,0.3333333emF'-F,6%Q\"yF'F/F2-F66$-F#6
#-F,6%Q\"xF'F/F2FAF=Fin-F>6-Q\"=F'FAFC/FGFEFHFJFLFNFP/FSQ,0.2777778emF
'/FVFao-F>6-Q*&uminus0;F'FAFCF_oFHFJFLFNFP/FSQ,0.2222222emF'/FVFgo-I#m
nGF$6$Q\"1F'FA-F>6-Q#..F'FAFCF_oFHFJFLFNFPFfo/FVFT-Fjo6$Q\"2F'FAF=-F66
&-F#6#-F66&-F#6-FX-F66$-F#6#-Fjo6$Q\"0F'FAFAF\\oFapF=-F,6%Q\"DF'/F0FEF
A-F66$-F#6#FXFAF\\qF\\oFco-Fjo6$Q#20F'FAFA/%%openGQ\"[F'/%&closeGQ\"]F
'FAF^rFarF=-F,6%Q*linecolorF'F/F2F\\o-F,6%Q%blueF'F/F2FA-F>6-Q\";F'FAF
CFFFHFJFLFNFPFRFbo" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 0 "" }}{PARA 0
 "" 0 "" {TEXT 208 148 "For another example, the general solution for \+
the equation y'' + x^2 y' + y = 0 can be found by Maple, but not using
 functions you might recognize! " }}}{EXCHG {PARA 0 "> " 0 "" 
{XPPEDIT 19 1 "`:=`(ode3, `+`(diff(y(x), `$`(x, 2)), `*`(`^`(x, 2), `*
`(diff(y(x), x))), y(x)) = 0);" "-I%mrowG6#/I+modulenameG6\"I,Typesett
ingGI(_syslibGF'61-I#miGF$6%Q%ode3F'/%'italicGQ%trueF'/%,mathvariantGQ
'italicF'-I#moGF$6-Q*≔F'/F3Q'normalF'/%&fenceGQ&falseF'/%*sepa
ratorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/
%'accentGF=/%'lspaceGQ,0.2777778emF'/%'rspaceGFL-F,6%Q%diffF'F/F2-I(mf
encedGF$6$-F#6(-F,6%Q\"yF'F/F2-FS6$-F#6#-F,6%Q\"xF'F/F2F9-F66-Q\",F'F9
F;/F?F1F@FBFDFFFH/FKQ&0.0emF'/FNQ,0.3333333emF'Fhn-F66-Q\"$F'F9F;F>F@F
BFDFFFHF_o/FNF`o-I#mnGF$6$Q\"2F'F9F9-F66-Q\"+F'F9F;F>F@FBFDFFFH/FKQ,0.
2222222emF'/FNF_p-FS6$-F#6#-I%msupGF$6%Fhn-F#6#Fgo/%1superscriptshiftG
Q\"0F'F9-F66-Q'⋅F'F9F;F>F@FBFDFFFHF_oFfoFO-FS6$-F#6&FWFZF[oFhnF9F
[pFWFZ-F66-Q\"=F'F9F;F>F@FBFDFFFHFJFM-Fho6$Q\"0F'F9-F66-Q\";F'F9F;F^oF
@FBFDFFFHF_oFM" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "dsolve(ode3
, y(x));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#m
iGF$6%Q'dsolveF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I(mfencedG
F$6$-F#6&-F,6%Q%ode3F'F/F2-I#moGF$6-Q\",F'/F3Q'normalF'/%&fenceGQ&fals
eF'/%*separatorGF1/%)stretchyGFE/%*symmetricGFE/%(largeopGFE/%.movable
limitsGFE/%'accentGFE/%'lspaceGQ&0.0emF'/%'rspaceGQ,0.3333333emF'-F,6%
Q\"yF'F/F2-F66$-F#6#-F,6%Q\"xF'F/F2FAFA-F>6-Q\";F'FAFCFFFHFJFLFNFPFR/F
VQ,0.2777778emF'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 147 "Let's graph
 the numerical solutions corresponding to a bunch of different initial
 conditions (IC), using two initial conditions for each solution. " }}
}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "`:=`(ic, \{seq(seq([y(0) = y0
, (D(y))(0) = k], y0 = -2 .. 2), k = -1 .. 1)\});" "-I%mrowG6#/I+modul
enameG6\"I,TypesettingGI(_syslibGF'6&-I#miGF$6%Q#icF'/%'italicGQ%trueF
'/%,mathvariantGQ'italicF'-I#moGF$6-Q*≔F'/F3Q'normalF'/%&fence
GQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.
movablelimitsGF=/%'accentGF=/%'lspaceGQ,0.2777778emF'/%'rspaceGFL-I(mf
encedGF$6&-F#6$-F,6%Q$seqF'F/F2-FP6$-F#6+FT-FP6$-F#6*-FP6&-F#6--F,6%Q
\"yF'F/F2-FP6$-F#6#-I#mnGF$6$Q\"0F'F9F9-F66-Q\"=F'F9F;F>F@FBFDFFFHFJFM
-F,6%Q#y0F'F/F2-F66-Q\",F'F9F;/F?F1F@FBFDFFFH/FKQ&0.0emF'/FNQ,0.333333
3emF'-F66-Q\"~F'F9F;F>F@FBFDFFFHFbp/FNFcp-F,6%Q\"DF'/F0F=F9-FP6$-F#6#F
]oF9F`oFho-F,6%Q\"kF'F/F2F9/%%openGQ\"[F'/%&closeGQ\"]F'F^pF[pFho-F66-
Q*&uminus0;F'F9F;F>F@FBFDFFFH/FKQ,0.2222222emF'/FNF_r-Feo6$Q\"2F'F9-F6
6-Q#..F'F9F;F>F@FBFDFFFHF^rFipFarF9F^pFbqFhoF[r-Feo6$Q\"1F'F9FdrFgrF9F
9/FfqQ\"|frF'/FiqQ\"|hrF'-F66-Q\";F'F9F;FapF@FBFDFFFHFbpFM" }}}{EXCHG 
{PARA 0 "> " 0 "" {XPPEDIT 19 1 "DEplot(ode3, y(x), x = -2 .. 3, ic, l
inecolor = x, thickness = 2);" "-I%mrowG6#/I+modulenameG6\"I,Typesetti
ngGI(_syslibGF'6%-I#miGF$6%Q'DEplotF'/%'italicGQ%trueF'/%,mathvariantG
Q'italicF'-I(mfencedGF$6$-F#68-F,6%Q%ode3F'F/F2-I#moGF$6-Q\",F'/F3Q'no
rmalF'/%&fenceGQ&falseF'/%*separatorGF1/%)stretchyGFE/%*symmetricGFE/%
(largeopGFE/%.movablelimitsGFE/%'accentGFE/%'lspaceGQ&0.0emF'/%'rspace
GQ,0.3333333emF'-F,6%Q\"yF'F/F2-F66$-F#6#-F,6%Q\"xF'F/F2FAF=Fin-F>6-Q
\"=F'FAFC/FGFEFHFJFLFNFP/FSQ,0.2777778emF'/FVFao-F>6-Q*&uminus0;F'FAFC
F_oFHFJFLFNFP/FSQ,0.2222222emF'/FVFgo-I#mnGF$6$Q\"2F'FA-F>6-Q#..F'FAFC
F_oFHFJFLFNFPFfo/FVFT-Fjo6$Q\"3F'FAF=-F,6%Q#icF'F/F2F=-F,6%Q*linecolor
F'F/F2F\\oFinF=-F>6-Q\"~F'FAFCF_oFHFJFLFNFPFRF`p-F,6%Q*thicknessF'F/F2
F\\oFioFA-F>6-Q\";F'FAFCFFFHFJFLFNFPFRFbo" }}}{EXCHG {PARA 0 "> " 0 ""
 {XPPEDIT 19 1 "" "%#%?G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 0 "" }}
{PARA 0 "" 0 "" {TEXT 208 17 "END OF WORKSHEET." }}{PARA 0 "" 0 "" 
{TEXT 208 72 "Worksheet created by Dr. A. Foster, Dept. of Mathematics
 and Statistics," }}{PARA 0 "" 0 "" {TEXT 208 60 "Memorial University \+
of Newfoundland, St. John's, NF, Canada." }}}}
{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2
 33 1 1 }