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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 11 "Math 2250-3" }}{PARA 257 "" 0 "
" {TEXT -1 20 "Wednesday October 29" }}{PARA 258 "" 0 "" {TEXT 259 0 "
" }{TEXT -1 21 "Solving DE's in Maple" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 190 "These notes discuss material from se
ctions 5.5 and 5.6 of the text, in the context of Maple.  They  are on
line at our Maple page, go to www.math.utah.edu/~korevaar/2250fall03/2
250maple.html." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 256 43 "Part 1: checking homework problems on Maple" }}{PARA 0 "
" 0 "" {TEXT -1 52 "     Here's a homework problem worked out in Maple
, " }{TEXT 260 20 "#12 from section 5.5" }{TEXT -1 127 ".  This proble
m was fairly painful to work by hand.  For more information on the com
mands which are used, use the Help files.  " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "with(DEtools):  #D
E command library" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "f:=x-
>2-sin(x):  #inhomogeneous term\ndeqtn:=diff(y(x),x,x,x)+ diff(y(x),x)
=f(x):\n   #the differential equation, #12 page 337." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 19 "dsolve(deqtn,y(x));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"yG6#%\"xG,.*&%$_C2G\"\"\"-%$cosGF&F+!\"\"*&%$_C1GF
+-%$sinGF&F+F+F,F+*&#F+\"\"#F+*&F'F+F1F+F+F+*&F5F+F'F+F+%$_C3GF+" }}}
{PARA 0 "" 0 "" {TEXT -1 102 "Discarding the pieces which solve the ho
mogeneous equation, the simplest particular solution we see is" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "yp:=x->1/2*x*sin(x)+2*x;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ypGf*6#%\"xG6\"6$%)operatorG%&arrow
GF(,&*&#\"\"\"\"\"#F/*&9$F/-%$sinG6#F2F/F/F/*&F0F/F2F/F/F(F(F(" }}}
{PARA 0 "" 0 "" {TEXT -1 298 "The method of guessing as modified for p
roblems when the guess for yp involves homogeneous equation solutions,
  would have had us to a guess of the form  yp=Ax + x(Bsin(x)+Ccos(x))
.  The algebra in computing L(yp) was messy.  We could check the algeb
ra part of our work as well as the final answer:" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 33 "yp:=x->A*x+x*B*sin(x)+x*C*cos(x);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#ypGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(*&%
\"AG\"\"\"9$F/F/*(F0F/%\"BGF/-%$sinG6#F0F/F/*(F0F/%\"CGF/-%$cosGF5F/F/
F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "diff(yp(x),x,x,x)
+diff(yp(x),x)=2-sin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(\"\"#
\"\"\"%\"BGF'-%$sinG6#%\"xGF'!\"\"*(F&F'%\"CGF'-%$cosGF+F'F-%\"AGF',&F
&F'F)F-" }}}{PARA 0 "" 0 "" {TEXT -1 101 "Equating coefficients we see
 that -2B=-1, C=0, and A=2.  This leads to yp=(1/2)xsin(x)+2x, as befo
re." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "   \+
  " }{TEXT 261 14 "How about #37?" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 246 "f:=x->1+x*exp(x):  #right hand side, then DE:\ndeqtn
:= diff(y(x),x,x,x) -2*diff(y(x),x,x) \n     + diff(y(x),x) =f(x);\nic
s:=y(0)=0, D(y)(0)=0, D(D(y))(0)=1:\n      #initial conditions...see h
elp files for syntax! \nans37:=dsolve(\{deqtn,ics\},y(x));\n  " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&deqtnG/,(-%%diffG6$-%\"yG6#%\"xG-%
\"$G6$F-\"\"$\"\"\"*&\"\"#F2-F(6$F*-F/6$F-F4F2!\"\"-F(6$F*F-F2,&F2F2*&
F-F2-%$expGF,F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ans37G/-%\"yG6
#%\"xG,.*&\"\"%\"\"\"-%$expGF(F-!\"\"*(\"\"$F-F)F-F.F-F-*&#F-\"\"'F-*&
)F)F2F-F.F-F-F-*&#F-\"\"#F-*&)F)F:F-F.F-F-F0F)F-F,F-" }}}{PAGEBK }
{PARA 0 "" 0 "" {TEXT 262 28 "Part 2:  Forced Oscillators:" }}{PARA 0 
"" 0 "" {TEXT 263 18 "Example 1 page 340" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 95 "with(plots):\nplot(5*cos(3*t)-5*cos(5*t),t=0..6*Pi,co
lor=black,\ntitle=`Figure 5.6.2 page 341`);\n" }}{PARA 13 "" 1 "" 
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ABELSG6$Q\"t6\"Q!Ffhr-%'COLOURG6&%$RGBGF)F)F)-%&TITLEG6#%6Figure~5.6.2
~page~341G-%%VIEWG6$;F($\"+#fb\\)=!\")%(DEFAULTG" 1 2 0 1 10 0 2 6 1 
4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 8 "Beating:" }}{PARA 0 "" 0 
"" {TEXT -1 390 "  If the forcing frequency is not equal to the natura
l frequency the general solution to will be the superposition of two c
os-sin terms, one corresponding to the particular solution with angula
r frequency w, and the other being the general solution to the homogen
eous problem, with angular frequency w0.  When w and w0 as well as the
 corresponding amplitudes are close, the system exhibits " }{TEXT 258 
7 "beating" }{TEXT -1 59 ".   Musicians tune their instruments using t
his phenomenon." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "restart:w
ith(DEtools):with(plots):\n   #I want to undefine m, x0 etc." }}{PARA 
7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefin
ed\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "deqtn1a:=diff(x(t)
,t,t) + w0^2*x(t) = (F0/m)*cos(w*t);\n   #this is the case w not equal
 to w0,\n   #I divided the model equation by m" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(deqtn1aG/,&-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"
\"\"*&)%#w0GF1F2F*F2F2*(%#F0GF2%\"mG!\"\"-%$cosG6#*&%\"wGF2F-F2F2" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "dsolve(deqtn1a,x(t));\n  #g
eneral solution, equal to particular\n  #solution plus general homogen
eous eqtn solution. " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG
,(*&-%$sinG6#*&%#w0G\"\"\"F'F/F/%$_C2GF/F/*&-%$cosGF,F/%$_C1GF/F/**%#F
0GF/-F36#*&%\"wGF/F'F/F/%\"mG!\"\",&*$)F.\"\"#F/F/*$)F:F@F/F<F<F/" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "sol4:=dsolve(\{deqtn1a,x(0)=
0,D(x)(0)=0\},x(t));\n  #a nice choice of initial conditions" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%sol4G/-%\"xG6#%\"tG,&**-%$cosG6#*&%#w0G\"
\"\"F)F1F1%#F0GF1%\"mG!\"\",&*$)F0\"\"#F1F4*$)%\"wGF8F1F1F4F1**F2F1-F-
6#*&F;F1F)F1F1F3F4,&F6F1F9F4F4F1" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 64 "sol5:=subs(\{w=45,w0=55,F0=50,m=.1\},sol4);\n   #Exam
ple 2 page 342" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol5G/-%\"xG6#%\"
tG,&*&$\"+++++]!#5\"\"\"-%$cosG6#,$*&\"#bF/F)F/F/F/!\"\"*&$\"+++++]F.F
/-F16#,$*&\"#XF/F)F/F/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
220 "actual:=plot(rhs(sol5),t=0..3, color=black):\nenvel1:=plot(sin(5*
t),t=0..3,color=black,linestyle=2):\nenvel2:=plot(-sin(5*t),t=0..3,col
or=black,linestyle=2):\ndisplay(\{actual, envel1,envel2\},title=`Figur
e 5.6.3 page 342`); " }}{PARA 13 "" 1 "" {GLPLOT2D 400 222 222 
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Fdcz$!3u@0_!*\\[!***F57$Fhdz$!3Ed&z/E>*****F57$F\\fz$!3s^:ZPum$***F57$
F`gz$!3hyLCG$R<(**F57$Fdhz$!35lC*\\P;5))*F57$Fhiz$!3r(yjvT>$G(*F57$Fh^
[l$!31</I]u&pE*F57$F`a[l$!3-iIT\"=4Hf)F57$Ffe[l$!3H-**4nYB\\wF57$F^h[l
$!3Do6d,%yG]'F5Fbh[lFaa]l-%&TITLEG6#%6Figure~5.6.3~page~342G-%+AXESLAB
ELSG6%Q\"t6\"Q!Ffj^l-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(F^h[lF[[_l" 1 2 
0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2
" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }{TEXT 257 63 " Resonance in undamped, forced harmonic oscilla
tors, when w=w0:" }{TEXT -1 221 "  You can use variation of parameters
 or undetermined coefficients to  solve the forced oscillator, with no
 damping, in the case that the driving frequency w exactly equals the \+
natural frequency w0.  Or, you can use Maple" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 89 "deqtn1:=diff(x(t),t,t) + w0^2*x(t) = (F0/m)*cos(w0
*t);\n  #I have again written k/m = w0^2" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'deqtn1G/,&-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&)%#w0
GF1F2F*F2F2*(%#F0GF2%\"mG!\"\"-%$cosG6#*&F5F2F-F2F2" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 47 "sol1:=dsolve(deqtn1,x(t));\n   #general s
olution" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol1G/-%\"xG6#%\"tG,(*&-
%$sinG6#*&%#w0G\"\"\"F)F1F1%$_C2GF1F1*&-%$cosGF.F1%$_C1GF1F1*&#F1\"\"#
F1**%#F0GF1,&F4F1*(F,F1F0F1F)F1F1F1%\"mG!\"\"F0!\"#F1F1" }}}{PARA 0 "
" 0 "" {TEXT -1 67 "You see there the particular solution which we fou
nd, namely xp(t):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "xp:=t->
(1/2)*F0*t*sin(w0*t)/(w0*m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xpG
f*6#%\"tG6\"6$%)operatorG%&arrowGF(,$*&#\"\"\"\"\"#F/*,%#F0GF/9$F/-%$s
inG6#*&%#w0GF/F3F/F/F8!\"\"%\"mGF9F/F/F(F(F(" }}}{PARA 0 "" 0 "" 
{TEXT -1 102 "You can see that xp(t)  solves the initial value problem
 x0=v0=0, i.e. the system initially at rest.  " }}{PARA 0 "" 0 "" 
{TEXT -1 73 "     For example, here's how to make the picture on page \+
343 of the text:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "F0:=100:
w0:=50:F0:=100:m:=1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "xp(t);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#*&%\"tG\"\"\"-%$sinG6#,$*&\"#]F%F$F%F%F%" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 63 "plot(xp(t),t=0..1.5,color=black,title=`Figure \+
5.6.4 page 343`);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 
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