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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 48 "Using Maple for Constant \+
Coefficient Linear ODEs" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 62 "First, define the _expression_ that I will be diff
erentiating." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f := exp(a*x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG-%$expG6#*&%\"aG\"\"\"%\"xGF*" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "Now define f1 to be the first d
erivative.  I name everything since I'll use the quantities again late
r." }}{PARA 0 "" 0 "" {TEXT -1 99 "Not that since I'm differentiating \+
an \"expression\" rather than a function, I use \"diff\".  Functions" 
}}{PARA 0 "" 0 "" {TEXT -1 29 "are differentiated using \"D\"." }
{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f1 := diff(f,x)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G*&%\"aG\"\"\"-%$expG6#*&F&F
'%\"xGF'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Define f2 to be the
 second derivative" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f2 := diff(f1
,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2G*&%\"aG\"\"#-%$expG6#*&F
&\"\"\"%\"xGF,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Define f3 to \+
be the third derivative" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f3 := di
ff(f2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f3G*&%\"aG\"\"$-%$expG
6#*&F&\"\"\"%\"xGF,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Define f
4 to be the fourth derivative." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f
4 := diff(f3,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f4G*&%\"aG\"\"%
-%$expG6#*&F&\"\"\"%\"xGF,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "T
o do problem 27 in section 3.3.  This is for the equation" }}{PARA 0 "
" 0 "" {TEXT -1 88 "y''' + 3 y'' + 3 y' + y = 0.  The expression expr \+
is the left hand side of the equation." }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 29 "expr := f3 + 3*f2 + 3*f1 + f;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%%exprG,**&%\"aG\"\"$-%$expG6#*&F'\"\"\"%\"xGF-F-F-*&F'\"\"#F)F-F(
*&F'F-F)F-F(F)F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "Now I want to
 divide exp(ax) out from the above.  To do this, I just rename expr, r
ather" }}{PARA 0 "" 0 "" {TEXT -1 37 "than introducing expr1, expr2, e
tc.  " }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "expr :=
 expr/exp(a*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG*&,**&%\"aG
\"\"$-%$expG6#*&F(\"\"\"%\"xGF.F.F.*&F(\"\"#F*F.F)*&F(F.F*F.F)F*F.F.F*
!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Force maple to factor ou
t the exp(ax):" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "expr := simplify(expr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%expr
G,**$%\"aG\"\"$\"\"\"*$F'\"\"#F(F'F(F)F)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 44 "Now factor the expression to find the roots." }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 21 "expr := factor(expr);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%exprG*$,&%\"aG\"\"\"F(F(\"\"$" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 91 "Now I've done all that I want maple to do.  Thinking
 for a moment, I see that all the roots" }}{PARA 0 "" 0 "" {TEXT -1 
100 "are repeated, so a set of linearly independet solutions will be y
1(x)  = exp(-x), y2(x) = x exp(-x)," }}{PARA 0 "" 0 "" {TEXT -1 74 "y3
(x) = x^2 exp(-x).  These can be found using the methods of section 3.
2." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
71 "I test these functions by defining _functions_ y1(x), y2(x), and y
3(x)." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "y1 := x
 -> exp(-x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G:6#%\"xG6\"6$%)o
peratorG%&arrowGF(-%$expG6#,$9$!\"\"F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "y2 := x -> x*exp(-x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#y2G:6#%\"xG6\"6$%)operatorG%&arrowGF(*&9$\"\"\"-%$expG6#,$F-!
\"\"F.F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "y3 := x -> x^
2*exp(-x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y3G:6#%\"xG6\"6$%)ope
ratorG%&arrowGF(*&9$\"\"#-%$expG6#,$F-!\"\"\"\"\"F(F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 80 "I test whether the functions satisfy the \+
ODE.  Note that I can use shorthand for" }}{PARA 0 "" 0 "" {TEXT -1 
77 "the higher derivatives: I write (D@@5)(y1)(x) instead of D(D(D(D(D
(y)))))(x)." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "(
D@@3)(y1)(x) + 3*(D@@2)(y1)(x) + 3*D(y1)(x) + y1(x);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "
(D@@3)(y2)(x) + 3*(D@@2)(y2)(x) + 3*D(y2)(x) + y2(x);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
53 "(D@@3)(y3)(x) + 3*(D@@2)(y3)(x) + 3*D(y3)(x) + y3(x);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "N
ow we'll ask maple to find the solutions from the very beginning.  Thi
s is a useful thing to " }}{PARA 0 "" 0 "" {TEXT -1 83 "know how to do
, except that you won't get credit on your homework or exams for just
" }}{PARA 0 "" 0 "" {TEXT -1 93 "writing down solutions that you appar
ently pulled out of thin air.  But it's good to know how" }}{PARA 0 "
" 0 "" {TEXT -1 9 "to do it." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "First I define the di
fferential equation I want to solve.  Note that y is a function, rathe
r than" }}{PARA 0 "" 0 "" {TEXT -1 44 "an expression, so D is used ins
tead of diff." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 
"diff_eqn := (D@@3)(y)(x) + 3*(D@@2)(y)(x) + 3*D(y)(x) + y(x) = 0;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)diff_eqnG/,*---%#@@G6$%\"DG\"\"$6#%
\"yG6#%\"xG\"\"\"---F*6$F,\"\"#F.F0F---F,F.F0F--F/F0F2\"\"!" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Use the dsolve command to find the
 solution.  Since no initial or boundary conditions are" }}{PARA 0 "" 
0 "" {TEXT -1 36 "specified it gives the general form." }{MPLTEXT 1 0 
0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "dsolve( diff_eqn,y(x));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&%$_C1G\"\"\"-%$expG6
#,$F'!\"\"F+F+*(%$_C2GF+F'F+F,F+F+*(%$_C3GF+F'\"\"#F,F+F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "I w
ant to do problem 35, which needs y'''''.  So I define f5:" }{MPLTEXT 
1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f5 := diff(f4,x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f5G*&%\"aG\"\"&-%$expG6#*&F&\"\"\"%
\"xGF,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Problem 35 asks for t
he general solution to" }}{PARA 0 "" 0 "" {TEXT -1 49 "y''''' + 5 y'''
' - 2 y''' - 10 y'' + y' + 5y = 0." }}{PARA 0 "" 0 "" {TEXT -1 58 "The
 expression expr is the left-hand side of the equation." }{MPLTEXT 1 
0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "expr := f5 + 5*f4 - 2*f3 -
 10*f2 + f1 + 5*f;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG,.*&%\"a
G\"\"&-%$expG6#*&F'\"\"\"%\"xGF-F-F-*&F'\"\"%F)F-F(*&F'\"\"$F)F-!\"#*&
F'\"\"#F)F-!#5*&F'F-F)F-F-F)F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 
"Proceeding as before," }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "expr := expr/exp(a*x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%%exprG*&,.*&%\"aG\"\"&-%$expG6#*&F(\"\"\"%\"xGF.F.F.*&F(\"\"%F
*F.F)*&F(\"\"$F*F.!\"#*&F(\"\"#F*F.!#5*&F(F.F*F.F.F*F)F.F*!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "expr := simplify(expr);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG,.*$%\"aG\"\"&\"\"\"*$F'\"\"%F
(*$F'\"\"$!\"#*$F'\"\"#!#5F'F)F(F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "factor(expr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&
%\"aG\"\"\"\"\"&F&F&,&F%F&!\"\"F&\"\"#,&F%F&F&F&F*" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 88 "So we have a family of 5 linearly independed solu
tions y1(x) = exp(-5x), y2(x) = exp(x)," }}{PARA 0 "" 0 "" {TEXT -1 
87 "y3(x) = x exp(x), y4(x) = exp(-x), y5(x) = x exp(-x).  Again, we c
an use dsolve to find" }}{PARA 0 "" 0 "" {TEXT -1 38 "the solutions.  \+
First, define the ODE:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 100 "diff_eqn := (D@@5)(y)(x) + 5*(D@@4)(y)(x) - 2*(D@@3)
(y)(x) - 10*(D@@2)(y)(x) + D(y)(x) + 5*y(x) = 0;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%)diff_eqnG/,.---%#@@G6$%\"DG\"\"&6#%\"yG6#%\"xG\"\"\"
---F*6$F,\"\"%F.F0F----F*6$F,\"\"$F.F0!\"#---F*6$F,\"\"#F.F0!#5--F,F.F
0F2-F/F0F-\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Find the gener
al solution:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "
dsolve( diff_eqn, y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%
\"xG,,*&%$_C1G\"\"\"-%$expGF&F+F+*&%$_C2GF+-F-6#,$F'!\"&F+F+*&%$_C3GF+
-F-6#,$F'!\"\"F+F+*(%$_C4GF+F,F+F'F+F+*(%$_C5GF+F6F+F'F+F+" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 88 "If I want to solve the initial value prob
lem, the initial values are defined as follows:" }{MPLTEXT 1 0 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "init_cond := y(0) = 1, D(y)(0) = 3,
 (D@@2)(y)(0) = 6, (D@@3)(y)(0) = - Pi, (D@@4)(y)(0) = 9 ;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%*init_condG6'/-%\"yG6#\"\"!\"\"\"/--%\"DG6
#F(F)\"\"$/---%#@@G6$F/\"\"#F0F)\"\"'/---F66$F/F1F0F),$%#PiG!\"\"/---F
66$F/\"\"%F0F)\"\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Give dsolv
e the ODE and the initial values.  It gives back an expression for y(x
)." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "dsolve( \{
diff_eqn, init_cond\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"y
G6#%\"xG*&,.*&,&#\"$0#\"#s\"\"\"%#PiG#F/\"\"%F/-%$expGF&\"\"#F/*&-F46#
,$F'!\"&F/F3F/#!\"\"\"$)G#!#f\"#KF/F0#F<F2*(,&#\"\"&\"#7F/F0FAF/F3F5F'
F/F/*&,&#!#<\"\")F/F0FAF/F'F/F/F/F3F<" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 90 "Now, here's something annoying about maple.  What you rea
lly would have liked would be for" }}{PARA 0 "" 0 "" {TEXT -1 97 "dsol
ve to give back something I could immediately plot.  But for some reas
on, if I ask it to plot" }}{PARA 0 "" 0 "" {TEXT -1 96 "y(x), it doesn
't seem to know what I'm talking about.  So I introduce a new expressi
on, z, which" }}{PARA 0 "" 0 "" {TEXT -1 62 "comes from taking the rig
ht-hand side of the above expression." }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 12 "z := rhs(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG*&,.*&,&#
\"$0#\"#s\"\"\"%#PiG#F,\"\"%F,-%$expG6#%\"xG\"\"#F,*&-F16#,$F3!\"&F,F0
F,#!\"\"\"$)G#!#f\"#KF,F-#F;F/*(,&#\"\"&\"#7F,F-F@F,F0F4F3F,F,*&,&#!#<
\"\")F,F-F@F,F3F,F,F,F0F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "We c
an now plot the solution z:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "plot(z,x=0...10);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'
CURVESG6$7`o7$\"\"!$\"\"\"F(7$$\"1nmm;arz@!#;$\"1I2cR<(=z\"!#:7$$\"1LL
$e9ui2%F.$\"1w+3V5:'p#F17$$\"1nmm\"z_\"4iF.$\"1#R78,P&[RF17$$\"1ommT&p
hN)F.$\"1d^YJb\\saF17$$\"1LLe*=)H\\5F1$\"1@OIlQ[ysF17$$\"1nm\"z/3uC\"F
1$\"17U=,]%pC*F17$$\"1++DJ$RDX\"F1$\"1XLY!GDK;\"!#97$$\"1nm\"zR'ok;F1$
\"1([8%f7p`9FP7$$\"1++D1J:w=F1$\"1Z<upo&fz\"FP7$$\"1MLL3En$4#F1$\"1Gd+
hH!Q@#FP7$$\"1nm;/RE&G#F1$\"1&)HQ+*Qwk#FP7$$\"1+++D.&4]#F1$\"1L`(o-\")
QA$FP7$$\"1+++vB_<FF1$\"1*yL]cPQ\"RFP7$$\"1+++v'Hi#HF1$\"16$z)fEg/ZFP7
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1MLLLY.KNF1$\"1_I4\"yt,$zFP7$$\"1++D\"o7Tv$F1$\"1R*fn(\\#zc*FP7$$\"1LL
L$Q*o]RF1$\"1Z\\i516G6!#87$$\"1,+D\"=lj;%F1$\"1s%Q6/_%\\8Fgq7$$\"1++vV
&R<P%F1$\"1B=h5^,)f\"Fgq7$$\"1ML$e9Ege%F1$\"1qH5,H;.>Fgq7$$\"1MLeR\"3G
y%F1$\"17By-C*4B#Fgq7$$\"1nm;/T1&*\\F1$\"1$p'=/!4Nk#Fgq7$$\"1nm\"zRQb@
&F1$\"1T%))y:&QYJFgq7$$\"1++v=>Y2aF1$\"1Qgm7,eaOFgq7$$\"1nm;zXu9cF1$\"
1a9I84!pG%Fgq7$$\"1+++]y))GeF1$\"1mK\"fASC/&Fgq7$$\"1++]i_QQgF1$\"1,%)
)\\,9S*eFgq7$$\"1,+D\"y%3TiF1$\"1Je5(*\\([$oFgq7$$\"1++]P![hY'F1$\"18G
['Hql-)Fgq7$$\"1LLL$Qx$omF1$\"1cM$)QLuQ#*Fgq7$$\"1+++v.I%)oF1$\"1YMGkR
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<F^v7$$\"1LLL3s?6zF1$\"1)o'pe9n^>F^v7$$\"1++DJXaE\")F1$\"1Ps_\"\\+?:#F
^v7$$\"1ommm*RRL)F1$\"1U]+l8jHBF^v7$$\"1om;a<.Y&)F1$\"16@=!\\qvZ#F^v7$
$\"1,]PM&*>^')F1$\"1N<Pj*G1`#F^v7$$\"1NLe9tOc()F1$\"1f/)4w&)\\c#F^v7$$
\"1,vV[ko/))F1$\"1ilQ25@tDF^v7$$\"1p;H#e0I&))F1$\"1KF2XV'fd#F^v7$$\"1^
(=#\\^;x))F1$\"1[F]g]6vDF^v7$$\"1Ne9;ZK,*)F1$\"1J0lPQosDF^v7$$\"1>H2$G
%[D*)F1$\"1[h$e\"QfoDF^v7$$\"1,++]Qk\\*)F1$\"1!=(G@`wiDF^v7$$\"1pmT5AS
g!*F1$\"1mG&Q7%R6DF^v7$$\"1NL$3dg6<*F1$\"1wci3C$GT#F^v7$$\"1,+voTAq#*F
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\"1N:dYc%)Q7F^v7$$\"1,++]oi\"o*F1$\"1i?+s)zW+\"F^v7$$\"1-]PMR<K(*F1$\"
1SA/Y'>[U(Fgq7$$\"1,+v=5s#y*F1$\"1n&f4fmh]%Fgq7$$\"1_iS\"*3))4)*F1$\"1
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*QXli)))FP7$$\"1,]P40O\"*)*F1$!1n&y(o%))=*GFgq7$$\"1]7.#Q?&=**F1$!1NBp
9Zy/]Fgq7$$\"1,voa-oX**F1$!12^v)pL=B(Fgq7$$\"1^PMF,%G(**F1$!1*4>[`>xd*
Fgq7$$\"#5F($!1sqRt(HZ?\"F^v-%'COLOURG6&%$RGBG$Ff`l!\"\"F(F(-%+AXESLAB
ELSG6$%\"xG%!G-%%VIEWG6$;F(Fe`l%(DEFAULTG" 2 250 250 250 2 0 1 0 2 9 
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{PARA 0 "" 0 "" {TEXT -1 110 "Now check that the solution satisfies th
e boundary conditions:  Note that in the following, if I don't want to
" }}{PARA 0 "" 0 "" {TEXT -1 61 "see the output, I use a : to finish t
he line, rather than a ;" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "subs(x=0,z): evalf(\");" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "z1 :=
 diff(z,x): subs(x=0,z1): evalf(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#$\"\"$\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "z2 := diff
(z1,x): subs(x=0,z2): evalf(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"\"'\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "z3 := diff(z2
,x): subs(x=0,z3): evalf(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+aE
fTJ!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "z4 := diff(z3,x)
: subs(x=0,z4): evalf(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"*\"
\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Now let's do a boundary va
lue problem.  I choose 5 boundary conditions.  We haven't talked" }}
{PARA 0 "" 0 "" {TEXT -1 97 "about boundary value problems yet, but it
's not automatic that all choices of boundary conditions" }}{PARA 0 "
" 0 "" {TEXT -1 85 "will be solvable.  I.e., there may be no solution \+
at all that satisfies the equation." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Forging ahead," }{MPLTEXT 1 0 0 "
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "bound_cond := y(0) = 1, D(y)(0)
 = 0, (D@@2)(y)(0) = Pi, y(2) = 3, D(y)(2) = 8;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%+bound_condG6'/-%\"yG6#\"\"!\"\"\"/--%\"DG6#F(F)F*/--
-%#@@G6$F/\"\"#F0F)%#PiG/-F(6#F7\"\"$/-F.F;\"\")" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 36 "dsolve(\{diff_eqn, bound_cond\},y(x));" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,,*(,8*$-%$expG6#\"\"#
\"\"%!#N*&F-\"\"'%#PiG\"\"\"\"#F*&F5F6F-F1F6*$F-F4\"$N#*$F-\"\"&!$k&*$
F-\"\"(\"$7\"\"#VF6F-!##*F5!\"**$F-F0\"#8*&F-F0F5F6FEF6,,!#8F6F9!#*)F,
\"\"*FD!\"(*$F-\"\")F1!\"\"-F.F&F0#FNF1*,F-F1,2F6F6FF!#=F5F6FD\"#9F-\"
$?\"*$F-\"\"$FMF,F6F8F6F6FGFN-F.6#,$F'!\"&F6FOF6#F6F1*&,8FJF6F-\"##*F5
FJFD\"#:F9\"$N\"FL!#:FFFHF3!#XF;\"$%oF>!$/\"*&F-FMF5F6F6F6FGFNFP**,8F-
!#YF5!#6F,F\\oFDFWF9\"#XF8FWFFFWF3F<F;!$A&F>\"#SFjnF6F6FGFNFOF0F'F6#F6
F0*(,8FLF<F`oF6F>\"#WF3FKF9\"#?F;\"#**FD!\"#FFF<F-!#BF6F6F5F6F6FGFNF'F
6F6F6FOFN" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Again, define the ex
prssion z using the right-hand side.  I've suppressed seeing the outpu
t." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "z = rhs(\"
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "plot(z,x=0...2);" }}
{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7S7$\"\"!$\"\"\"F(7$$\"1LLLL3
VfV!#<$\"1AfJ\"HUk8\"!#:7$$\"1nmm\"H[D:)F.$\"1jq#=_[UE\"F17$$\"1LLLe0$
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q(G**y\"F1$\"1N/)z<E%\\;Fhu7$$\"1nm;9@BM=F1$\"1Q]%yYeMs\"Fhu7$$\"1LLL`
v&Q(=F1$\"1UC%e(Q#>z\"Fhu7$$\"1++DOl5;>F1$\"1\"4lWMZt'=Fhu7$$\"1++v.Ua
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0 0 0 0 0 0 0 0 1 1 0 0 0 2 105 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 101 "The above plot makes me nervous.  The solution was suppo
sed to equal 3 at x = 2.  Let's check whether" }}{PARA 0 "" 0 "" 
{TEXT -1 49 "the solution satisfies the boundary conditions..." }
{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "subs(x=0,z): ev
alf(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"\"\"\"!" }}}{EXCHG 
{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "z
1 := diff(z,x) : subs(x=0,z1): evalf(\");" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"\"$\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
41 "z2 := diff(z1,x): subs(x=0,z2): evalf(\");" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"\"'\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "subs(x=2,z): evalf(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+e/*
[-#!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(x=2,z1): ev
alf(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+a*f<%>!\")" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 103 "The solution maple gave us doesn't satis
fy the boundary conditions.  The above numbers are supposed to " }}
{PARA 0 "" 0 "" {TEXT -1 25 "equal 1, 0, pi, 3, and 8." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "What's going on?  I
s the problem that there is no solution and maple felt obliged to give
 an answer?  Is it" }}{PARA 0 "" 0 "" {TEXT -1 109 "that there is a so
lution but for some reason maple couldn't find it?  Keep tuned!  And u
ntil then, be careful" }}{PARA 0 "" 0 "" {TEXT -1 28 "with blindly tru
sting maple." }}}}{MARK "0 0 0" 48 }{VIEWOPTS 1 1 0 1 1 1803 }