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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 40 "Partial Differential Equa
tions PowerTool" }}{PARA 19 "" 0 "" {TEXT -1 16 "by Dr. Jim Herod" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 301 51 "Section \+
3.1: Ordinary Differential Equations Review" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 302 29 "Maple Packages in Sect
ion 3.1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 216 "    In order that ideas from \+
ordinary differential equations will be fresh when we begin a discussi
on of partial differential equations, we review pertinent ideas from o
rdinary differential equations in this Section." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "We make our review by ask
ing six questions." }}{PARA 0 "" 0 "" {TEXT 256 11 "Question 1." }
{TEXT -1 9 " Suppose " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 
-1 76 " > 0. Which of these is a pair of linearly independent solution
s for Y '' - " }{XPPEDIT 18 0 "lambda^2;" "6#*$%'lambdaG\"\"#" }{TEXT 
-1 14 " Y = 0 on [0, " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 2 "]?
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "A. exp
(" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 14 " x) and exp( -
" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 18 " x),      C. si
nh(" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 14 " x ) and cos
h(" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 4 " x)," }}{PARA 
0 "" 0 "" {TEXT -1 7 "B. sin(" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" 
}{TEXT -1 12 " x) and cos(" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }
{TEXT -1 22 " x ),         D. sinh(" }{XPPEDIT 18 0 "lambda;" "6#%'lam
bdaG" }{TEXT -1 15 " x ) and sinh( " }{XPPEDIT 18 0 "lambda;" "6#%'lam
bdaG" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 7 " - \+
x))." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 
274 18 "Answer Using Maple" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 264 16 "Re
garding Part A" }}{PARA 0 "" 0 "" {TEXT -1 19 "The functions exp( " }
{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 1 " " }{TEXT 262 1 "x
" }{TEXT -1 11 ") and exp( " }{XPPEDIT 18 0 "-lambda;" "6#,$%'lambdaG!
\"\"" }{TEXT -1 1 " " }{TEXT 263 1 "x" }{TEXT -1 87 ") are linearly in
dependent. We ask if they are solutions for the differential equation.
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Y1:=x->exp(lambda*x);\nY
2:=x->exp(-lambda*x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "di
ff(Y1(x),x,x)-lambda^2*Y1(x);\ndiff(Y2(x),x,x)-lambda^2*Y2(x);" }}}
{PARA 0 "" 0 "" {TEXT -1 60 "We see that these functions solve the dif
ferential equation." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 267 16 "Regardin
g Part B" }}{PARA 0 "" 0 "" {TEXT -1 19 "The functions sin( " }
{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 1 " " }{TEXT 265 1 "x
" }{TEXT -1 11 ") and cos( " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }
{TEXT -1 1 " " }{TEXT 266 1 "x" }{TEXT -1 87 ") are linearly independe
nt. We ask if they are solutions for the differential equation." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Y1:=x->sin(lambda*x);\nY2:=x
->cos(lambda*x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "diff(Y1
(x),x,x)-lambda^2*Y1(x);\ndiff(Y2(x),x,x)-lambda^2*Y2(x);" }}}{PARA 0 
"" 0 "" {TEXT -1 67 "We see that these functions do not solve the diff
erential equation." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 270 16 "Regarding
 Part C" }}{PARA 0 "" 0 "" {TEXT -1 20 "The functions sinh( " }
{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 1 " " }{TEXT 268 1 "x
" }{TEXT -1 12 ") and cosh( " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }
{TEXT -1 1 " " }{TEXT 269 1 "x" }{TEXT -1 87 ") are linearly independe
nt. We ask if they are solutions for the differential equation." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Y1:=x->sinh(lambda*x);\nY2:=
x->cosh(lambda*x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "diff(
Y1(x),x,x)-lambda^2*Y1(x);\ndiff(Y2(x),x,x)-lambda^2*Y2(x);" }}}{PARA 
0 "" 0 "" {TEXT -1 60 "We see that these functions solve the different
ial equation." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 272 16 "Regarding Part
 D" }}{PARA 0 "" 0 "" {TEXT -1 20 "The functions sinh( " }{XPPEDIT 18 
0 "lambda;" "6#%'lambdaG" }{TEXT -1 1 " " }{TEXT 271 1 "x" }{TEXT -1 
12 ") and sinh( " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 3 "
 ( " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 3 " - " }{TEXT 273 1 "x
" }{TEXT -1 84 ")  linearly independent. We ask if they are solutions \+
for the differential equation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 50 "Y1:=x->sinh(lambda*x);\nY2:=x->sinh(lambda*(Pi-x));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "diff(Y1(x),x,x)-lambda^2*Y1(x);\ndi
ff(Y2(x),x,x)-lambda^2*Y2(x);" }}}{PARA 0 "" 0 "" {TEXT -1 261 "We see
 that these functions solve the differential equation. In time, we wil
l see that an advantage for this pair of solutions of the differential
 equation not shared by the other two pair of solutions is that one fu
nction is zero at 0 and the other is zero at " }{XPPEDIT 18 0 "Pi;" "6
#%#PiG" }{TEXT -1 1 "." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 288 9 "Ask Ma
ple" }}{PARA 0 "" 0 "" {TEXT -1 41 "Let's see what solutions Maple wil
l give." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "dsolve(diff(Y(x),
x,x)-lambda^2*Y(x)=0,Y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
59 "dsolve(diff(Y(x),x,x)-lambda^2*Y(x)=0,Y(x),method=laplace);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 56 "What solutions you get depends on what methods are used." }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 11 "Question
 2." }{TEXT -1 9 " Suppose " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }
{TEXT -1 76 " > 0. Which of these is a pair of linearly independent so
lutions for Y '' + " }{XPPEDIT 18 0 "lambda^2;" "6#*$%'lambdaG\"\"#" }
{TEXT -1 14 " Y = 0 on [0, " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 
2 "]?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "A
. exp(" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 14 " x) and e
xp( -" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 18 " x),      \+
C. sinh(" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 14 " x ) an
d cosh(" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 4 " x)," }}
{PARA 0 "" 0 "" {TEXT -1 7 "B. sin(" }{XPPEDIT 18 0 "lambda;" "6#%'lam
bdaG" }{TEXT -1 12 " x) and cos(" }{XPPEDIT 18 0 "lambda;" "6#%'lambda
G" }{TEXT -1 21 " x ),         D. sin(" }{XPPEDIT 18 0 "lambda;" "6#%'
lambdaG" }{TEXT -1 14 " x ) and sin( " }{XPPEDIT 18 0 "lambda;" "6#%'l
ambdaG" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 7 " \+
- x))." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" 
{TEXT 287 18 "Answer Using Maple" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 277 
16 "Regarding Part A" }}{PARA 0 "" 0 "" {TEXT -1 19 "The functions exp
( " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 1 " " }{TEXT 275 
1 "x" }{TEXT -1 11 ") and exp( " }{XPPEDIT 18 0 "-lambda;" "6#,$%'lamb
daG!\"\"" }{TEXT -1 1 " " }{TEXT 276 1 "x" }{TEXT -1 87 ") are linearl
y independent. We ask if they are solutions for the differential equat
ion." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Y1:=x->exp(lambda*x)
;\nY2:=x->exp(-lambda*x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
63 "diff(Y1(x),x,x)+lambda^2*Y1(x);\ndiff(Y2(x),x,x)+lambda^2*Y2(x);" 
}}}{PARA 0 "" 0 "" {TEXT -1 67 "We see that these functions do not sol
ve the differential equation." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 280 
16 "Regarding Part B" }}{PARA 0 "" 0 "" {TEXT -1 19 "The functions sin
( " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 1 " " }{TEXT 278 
1 "x" }{TEXT -1 11 ") and cos( " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG
" }{TEXT -1 1 " " }{TEXT 279 1 "x" }{TEXT -1 87 ") are linearly indepe
ndent. We ask if they are solutions for the differential equation." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Y1:=x->sin(lambda*x);\nY2:=x
->cos(lambda*x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "diff(Y1
(x),x,x)+lambda^2*Y1(x);\ndiff(Y2(x),x,x)+lambda^2*Y2(x);" }}}{PARA 0 
"" 0 "" {TEXT -1 63 "We see that these functions do solve the differen
tial equation." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 283 16 "Regarding Par
t C" }}{PARA 0 "" 0 "" {TEXT -1 20 "The functions sinh( " }{XPPEDIT 
18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 1 " " }{TEXT 281 1 "x" }{TEXT 
-1 12 ") and cosh( " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 
1 " " }{TEXT 282 1 "x" }{TEXT -1 87 ") are linearly independent. We as
k if they are solutions for the differential equation." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Y1:=x->sinh(lambda*x);\nY2:=x->cosh
(lambda*x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "diff(Y1(x),x
,x)+lambda^2*Y1(x);\ndiff(Y2(x),x,x)+lambda^2*Y2(x);" }}}{PARA 0 "" 0 
"" {TEXT -1 67 "We see that these functions do not solve the different
ial equation." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 285 16 "Regarding Part
 D" }}{PARA 0 "" 0 "" {TEXT -1 19 "The functions sin( " }{XPPEDIT 18 
0 "lambda;" "6#%'lambdaG" }{TEXT -1 1 " " }{TEXT 284 1 "x" }{TEXT -1 
11 ") and sin( " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 3 " \+
( " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 3 " - " }{TEXT 286 1 "x" 
}{TEXT -1 84 ")  linearly independent. We ask if they are solutions fo
r the differential equation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
48 "Y1:=x->sin(lambda*x);\nY2:=x->sin(lambda*(Pi-x));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "diff(Y1(x),x,x)+lambda^2*Y1(x);\ndi
ff(Y2(x),x,x)+lambda^2*Y2(x);" }}}{PARA 0 "" 0 "" {TEXT -1 263 "We see
 that these functions solve the differential equation. In Question 5, \+
we will see that an advantage for this pair of solutions of the differ
ential equation not shared by the other pair of solutions is that one \+
function is zero at 0 and the other is zero at " }{XPPEDIT 18 0 "Pi;" 
"6#%#PiG" }{TEXT -1 1 "." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 289 9 "Ask \+
Maple" }}{PARA 0 "" 0 "" {TEXT -1 40 "Let's see what solutions Maple w
ill give" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "dsolve(diff(Y(x)
,x,x)+lambda^2*Y(x)=0,Y(x),output=basis);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 258 11 "Question 3." }{TEXT -1 9 " Suppose " }{XPPEDIT 18 0 "lam
bda;" "6#%'lambdaG" }{TEXT -1 54 " > 0. Which of these is a bounded so
lution for Y '' - " }{XPPEDIT 18 0 "lambda^2;" "6#*$%'lambdaG\"\"#" }
{TEXT -1 14 " Y = 0 on [0, " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG
" }{TEXT -1 2 ")?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 7 "A. exp(" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 
15 " x)    B. exp(-" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 
16 " x)     C. sinh(" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 
-1 17 " x)      D. cosh(" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }
{TEXT -1 3 " x)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 94 "There are two issues here: which is a solution and which \+
is bounded on the specified interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 290 18 "Answer Using Maple" }}{PARA 
0 "" 0 "" {TEXT -1 143 "We have already seen that each of these is a s
olution for the differential equation. The only question that remains \+
is which is bounded. Using " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }
{TEXT -1 82 " = 1, we sketch all the graph in colors red, blue, green,
 and black, respectively." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 
"plot([exp(x),exp(-x),sinh(x),cosh(x)],x=0..2,\n            color=[red
,blue,green,black]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 259 11 "Question 4." }{TEXT -1 89 " Which of the
se is a bounded solution on the interval [0, 5] of the differential eq
uation" }}{PARA 0 "" 0 "" {TEXT -1 10 "          " }{XPPEDIT 18 0 "r^2
;" "6#*$%\"rG\"\"#" }{TEXT -1 33 " R ''(r) + r R '(r) - 9 R(r) = 0?" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "A. exp(3
 r)       B. " }{XPPEDIT 18 0 "r^3;" "6#*$%\"rG\"\"$" }{TEXT -1 49 "  \+
     C. sin(3 r)       D. exp(-3 r)       E. 1/" }{XPPEDIT 18 0 "r^3;
" "6#*$%\"rG\"\"$" }{TEXT -1 19 "       F. cosh(3 r)" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 291 18 "Answer Using M
aple" }}{PARA 0 "" 0 "" {TEXT -1 79 "It is easy enough to find what ar
e the solutions for the differential equation." }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 81 "dsolve(r^2*diff(R(r),r,r)+r*diff(R(r),r)-9*R(r)=
0,R(r),\n           output=basis);" }}}{PARA 0 "" 0 "" {TEXT -1 97 "We
 have left only to check which of these is bounded on [0,1]. Is this c
lear? We plot both these." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 
"plot([1/r^3,r^3],r=0..1,R=0..5,color=[black,red]);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 260 11 "Question 5." }{TEXT -1 14 " If  u(x,y) = " }
{XPPEDIT 18 0 "sum(a[p]*sin(p*x)*sinh(p*y));" "6#-%$sumG6#*(&%\"aG6#%
\"pG\"\"\"-%$sinG6#*&F*F+%\"xGF+F+-%%sinhG6#*&F*F+%\"yGF+F+" }{TEXT 
-1 3 " + " }{XPPEDIT 18 0 "sum(b[p]*sin(p*x)*sinh(p*(Pi-y)));" "6#-%$s
umG6#*(&%\"bG6#%\"pG\"\"\"-%$sinG6#*&F*F+%\"xGF+F+-%%sinhG6#*&F*F+,&%#
PiGF+%\"yG!\"\"F+F+" }{TEXT -1 6 "  and " }}{PARA 0 "" 0 "" {TEXT -1 
43 "                           u(x,0) = 0, u(x," }{XPPEDIT 18 0 "pi;" 
"6#%#piG" }{TEXT -1 13 ") = sin(2 x) " }}{PARA 0 "" 0 "" {TEXT -1 13 "
what are the " }{XPPEDIT 18 0 "a[p];" "6#&%\"aG6#%\"pG" }{TEXT -1 13 "
 's   and    " }{XPPEDIT 18 0 "b[p];" "6#&%\"bG6#%\"pG" }{TEXT -1 4 " \+
's?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 
299 18 "Answer Using Maple" }}{PARA 0 "" 0 "" {TEXT -1 54 "We define t
he function u as specified in the question." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 116 "u:=(x,y)->sum(a[p]*sin(p*x)*sinh(p*y),p=1..infinit
y)+ \n             sum(b[p]*sin(p*x)*sinh(p*(Pi-y)),p=1..infinity);" }
}}{PARA 0 "" 0 "" {TEXT -1 34 "We have two pieces of information." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "0=u(x,0);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 17 "sin(2*x)=u(x,Pi);" }}}{PARA 0 "" 0 "" 
{TEXT -1 24 "Before we determine the " }{TEXT 292 1 "a" }{TEXT -1 7 "'
s and " }{TEXT 293 1 "b" }{TEXT -1 40 "'s, note how convenient it was \+
that sin(" }{TEXT 294 3 "p y" }{TEXT -1 7 ")=0 at " }{TEXT 303 1 "y" }
{TEXT -1 19 " = 0 and that sin( " }{TEXT 295 1 "p" }{TEXT -1 2 " (" }
{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 1 " " }{TEXT 296 2 "-y" }
{TEXT -1 8 "))=0 at " }{TEXT 298 1 "y" }{TEXT -1 3 " = " }{XPPEDIT 18 
0 "Pi;" "6#%#PiG" }{TEXT -1 3 ". \n" }}{PARA 0 "" 0 "" {TEXT -1 119 "A
s you can see by looking at the next to last equation above, the first
 two pieces of information implies that all the " }{TEXT 297 0 "" }
{TEXT -1 0 "" }{XPPEDIT 18 0 "b[p];" "6#&%\"bG6#%\"pG" }{TEXT -1 92 "'
s are zero, and the second piece gives the last equation above, which \+
implies that all the " }{XPPEDIT 18 0 "a[p];" "6#&%\"aG6#%\"pG" }
{TEXT -1 36 "'s is zero except the second one -- " }{TEXT 304 1 "p" }
{TEXT -1 12 " = 2 -- and " }{XPPEDIT 18 0 "a[2];" "6#&%\"aG6#\"\"#" }
{TEXT -1 11 " =  1/sinh(" }{XPPEDIT 18 0 "2*pi;" "6#*&\"\"#\"\"\"%#piG
F%" }{TEXT -1 30 "). Thus here is a graph for u." }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 69 "plot3d(1/sinh(2*Pi)*sin(2*x)*sinh(2*y), x=0..P
i,y=0..Pi,axes=NORMAL);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 11 "Qu
estion 6." }{TEXT -1 9 " If u(r, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG
" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "sum(a[p]*sin(p*theta)*r^p,p);" "6
#-%$sumG6$*(&%\"aG6#%\"pG\"\"\"-%$sinG6#*&F*F+%&thetaGF+F+)%\"rGF*F+F*
" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "sum(b[p]*cos(p*theta)*r^p,p);" "6#
-%$sumG6$*(&%\"bG6#%\"pG\"\"\"-%$cosG6#*&F*F+%&thetaGF+F+)%\"rGF*F+F*
" }{TEXT -1 6 "  and " }}{PARA 0 "" 0 "" {TEXT -1 10 "     u(1, " }
{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 16 ") = 1 + 3 cos(3 " }
{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 13 ") + 5 sin( 2 " }
{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 2 ") " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "then what is u(r, " }
{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 22 "), u(0,0), and u(1/2
, " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 4 "/4)?" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 300 18 "Answer Using M
aple" }}{PARA 0 "" 0 "" {TEXT -1 67 "This problem is similar to the on
e above. First, define u as above." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 117 "u:=(r,theta)->sum(a[p]*sin(p*theta)*r^p,p=1..infinit
y) + \n                  sum(b[p]*cos(p*theta)*r^p,p=0..infinity);" }}
}{PARA 0 "" 0 "" {TEXT -1 40 "The information that is given specifies \+
" }{TEXT 305 1 "u" }{TEXT -1 6 " when " }{TEXT 306 1 "r" }{TEXT -1 5 "
 = 1." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "1+3*cos(3*theta)+5*
sin(2*theta)=u(1,theta);" }}}{PARA 0 "" 0 "" {TEXT -1 22 "This implies
 that all " }{XPPEDIT 18 0 "a[p];" "6#&%\"aG6#%\"pG" }{TEXT -1 7 "'s a
nd " }{XPPEDIT 18 0 "b[p];" "6#&%\"bG6#%\"pG" }{TEXT -1 19 "'s are zer
o except " }{XPPEDIT 18 0 "b[0];" "6#&%\"bG6#\"\"!" }{TEXT -1 6 " = 1,
 " }{XPPEDIT 18 0 "b[3];" "6#&%\"bG6#\"\"$" }{TEXT -1 10 " = 3, and " 
}{XPPEDIT 18 0 "a[2];" "6#&%\"aG6#\"\"#" }{TEXT -1 6 " = 5. " }}{PARA 
0 "" 0 "" {TEXT -1 8 "We give " }{TEXT 307 1 "u" }{TEXT -1 1 "(" }
{TEXT 308 1 "r" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" 
}{TEXT -1 3 "), " }{TEXT 309 1 "u" }{TEXT -1 18 "(0,0), and u(1/2, " }
{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 4 "/4)." }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 54 "u:=(r,theta)->5*sin(2*theta)*r^2+1+3*cos(3*the
ta)*r^3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "u(0,0);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "u(1/2,Pi/4);" }}}{PARA 0 "" 
0 "" {TEXT -1 29 "Finally, we sketch the graph." }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 182 "plot3d([r,theta,u(r,theta)],r=0..1,theta=-Pi.
.Pi,\n             coords=cylindrical,axes=normal,orientation=[20,40],
\n             numpoints=2000, lightmodel=light1, style=patchnogrid);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 271 "In this Section 3.1, we have looked at some of the simple diff
erential equations that will arise in obtaining solutions for partial \+
differential equations. In getting solutions, one sometimes must choos
e among all the solutions available to make the computations easier. \+
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 50 "EMAIL: herod@math.gatech.edu   or   jherod@tds.net" 
}}{PARA 0 "" 0 "" {TEXT -1 38 "URL: http://www.math.gatech.edu/~herod
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 36 "Cop
yright \251  2003  by James V. Herod" }}{PARA 256 "" 0 "" {TEXT -1 19 
"All rights reserved" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 
1 1 1 }{PAGENUMBERS 1 1 2 33 1 1 }