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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 11 "Math 2250-3" }}{PARA 257 "" 0 "
" {TEXT -1 17 "Friday November 5" }}{PARA 258 "" 0 "" {TEXT 259 0 "" }
{TEXT -1 21 "Solving DE's in Maple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 199 "These notes discuss material from sectio
ns 5.5 and 5.6 of the text, in the context of Maple.  They  are online
 at our class notes page, go to www.math.utah.edu/~korevaar/2250fall04
/2250lectures.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 Ma
ple, " }{TEXT 260 20 "#12 from section 5.5" }{TEXT -1 171 ".  This pro
blem is fairly painful to work by hand, as you will disover, if you ha
ven't already.  For more information on the commands which are used, u
se the Help files.  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "with
(DEtools):  #DE 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 346." }}
}{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+%$_C3GF+" }}}{PARA 0 "" 0 "" {TEXT -1 102 "Discarding the pieces wh
ich solve the homogeneous equation, the simplest particular solution w
e see is" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "yp:=x->1/2*x*sin
(x)+2*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ypGf*6#%\"xG6\"6$%)oper
atorG%&arrowGF(,&*&#\"\"\"\"\"#F/*&9$F/-%$sinG6#F2F/F/F/*&F0F/F2F/F/F(
F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 260 "The method of guessing would le
ad us to a guess of the form  yp=Ax + x(Bsin(x)+Ccos(x)), after accoun
ting for the homogeneous solutions.  The algebra in computing L(yp) by
 hand is messy.  We could check the algebra part of our work as well a
s 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#>%#ypG
f*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 before." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "     " }{TEXT 261 55 "How \+
about #37?  We can check the answer you should get:" }}{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);\nics:=y(0)=0, D(y)(0)=0, D(D(y))(0)=1:\n      #initial conditio
ns...see help 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 350" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 82 "plot(5*cos(3*t)-5*cos(5*t),t=0..6*Pi,color=black,\nti
tle=`Figure 5.6.2 page 350`);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 378 176 
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$\"+#fb\\)=!\")%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT 264 8 "Beating:" }}{PARA 0 "" 0 "" {TEXT -1 387 "  If the
 forcing frequency is not equal to the natural frequency the general s
olution will be the superposition of two cos-sin terms, one correspond
ing to the particular solution with angular frequency w, and the other
 being the general solution to the homogeneous problem, with angular f
requency w0.  When w and w0 as well as the corresponding amplitudes ar
e close, the system exhibits " }{TEXT 258 7 "beating" }{TEXT -1 59 ". \+
  Musicians tune their instruments using this phenomenon." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "restart:with(DEtools):with(plots):
\n   #I want to undefine m, x0 etc." }}{PARA 7 "" 1 "" {TEXT -1 50 "Wa
rning, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 132 "deqtn1a:=diff(x(t),t,t) + w0^2*x(t) = (F0/m
)*cos(w*t);\n   #this is the case w not equal to w0,\n   #I divided th
e model equation by m" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(deqtn1aG/,
&-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&)%#w0GF1F2F*F2F2*(%#F0G
F2%\"mG!\"\"-%$cosG6#*&%\"wGF2F-F2F2" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 115 "dsolve(deqtn1a,x(t));\n  #general solution, equal to
 particular\n  #solution plus general homogeneous eqtn solution. " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,(*&-%$sinG6#*&%#w0G\"\"
\"F'F/F/%$_C2GF/F/*&-%$cosGF,F/%$_C1GF/F/**%#F0GF/-F36#*&%\"wGF/F'F/F/
%\"mG!\"\",&*$)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)F1F1F3F4F5F4F
4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "sol5:=subs(\{w=45,w0=5
5,F0=50,m=.1\},sol4);\n   #Example 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,color=black,linestyle=2):\ndisplay(
\{actual, envel1,envel2\},title=`Figure 5.6.3 page 351`); " }}{PARA 
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F07$$\"3Tv$4Y6cb(HFez$!36Ko&HZ$p_mF07$$\"3b7y]&4I'zHFez$!3[:#GQ%[#o-(F
07$$\"38]iSwSq$)HFez$!3oAq+*y.Q4(F07$$\"3r(o/t0yx)HFez$!3r)y^2bCA'oF07
$$\"3%[7.#Q?&=*HFez$!3ej8[9ys`jF07$$\"3Ui:5>g#f*HFez$!3l\\:V*[\"e,cF07
$F^dm$!3#e!y-%\\a([YF0Fbdm-%&TITLEG6#%6Figure~5.6.3~page~351G-%+AXESLA
BELSG6%Q\"t6\"Q!Ffj^l-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(F^dmF[[_l" 1 2 
0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2
" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }{TEXT 257 63 " Resonance in undamped, forced harmonic oscilla
tors, when w=w0:" }{TEXT -1 221 "  You can use variation of parameters
 or undetermined coefficients to  solve the forced oscillator, with no
 damping, in the case that the driving frequency w exactly equals the \+
natural frequency w0.  Or, you can use Maple" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 89 "deqtn1:=diff(x(t),t,t) + w0^2*x(t) = (F0/m)*cos(w0
*t);\n  #I have again written k/m = w0^2" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'deqtn1G/,&-%%diffG6$-%\"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%$_C1GF1F1*&#F1\"\"#
F1*,%#F0GF1F,F1F0!\"\"F)F1%\"mGF<F1F1" }}}{PARA 0 "" 0 "" {TEXT -1 67 
"You see there the particular solution which we found, namely xp(t):" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "xp:=t->(1/2)*F0*t*sin(w0*t
)/(w0*m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xpGf*6#%\"tG6\"6$%)ope
ratorG%&arrowGF(,$*&#\"\"\"\"\"#F/*,%#F0GF/9$F/-%$sinG6#*&%#w0GF/F3F/F
/F8!\"\"%\"mGF9F/F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 102 "You can se
e that xp(t)  solves the initial value problem x0=v0=0, i.e. the syste
m initially at rest.  " }}{PARA 0 "" 0 "" {TEXT -1 73 "     For exampl
e, here's how to make the picture on page 352 of the text:" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "F0:=100:w0:=50:F0:=100:m:=1:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "xp(t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&%\"tG\"\"\"-%$sinG6#,$*&\"#]F%F$F%F%F%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "plot(xp(t),t=0..1.5,color=black,tit
le=`Figure 5.6.4 page 352`);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 
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