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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 132 "Worksheet for Column Pr
oject - Exact Solution:  Corrected Using Change of Variables To Avoid \+
MapleVrelease5 Bug with Airy Functions." }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Order :=
 8;" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 32 "Define the differential \+
equation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "eqn3 := diff(u(x),x$2) \+
+ a^2*x*u(x)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "yt := x-
> 2*a*x^(3/2)/3; vt :=x-> u(x)/sqrt(x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 62 "xt := y-> (3*y/(2*a))^(2/3); ut := y-> (3*y/(2*a))^(1
/3)*v(y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Dyt := diff(yt
(x),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Dy := simplify(s
ubs(x=(3*y/(2*a))^(2/3),a*sqrt(x)));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 69 "eqnt := (2*y/(3*a))*(diff(diff(ut(y),y)*Dy,y)*Dy+a^2*
xt(y)*ut(y)) =0;" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 38 "Solve the O
DE with two conditions at 0" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "soln
t := dsolve(eqnt,v(y));" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 36 "Assi
gn the solution to the name u(x)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 
"assign(solnt); V := unapply(v(y),y);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "U :=x-> sqrt(x)*V(yt(x));" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 23 "U0 := limit(U(x),x=0); " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 21 "_C2:=solve(U0=1,_C2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "DU := diff(U(x),x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "DU0 := limit(DU,x=0);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "_C1 := solve(DU0=0,_C1);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "solnU := U(x);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 
-1 137 "Notice that the solution is in terms of Bessel functions.  Doe
s this make\nsense since Bessel's eqn has a regular singular point at \+
x=0?\n\n" }}{PARA 256 "" 0 "" {TEXT -1 97 "How does this series expans
ion of the exact solution compare with\nthe solution you constructed?
\n\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "solnUseries := series(U(x),
x=0,8);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 33 "Evaluate the constan
t a for r=0.5" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "val := a = sqrt(4*
(0.28)/(2.8*10^7)/0.5^2);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 57 "Lo
cate the root - give a search range to help the routine" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 50 "ans3inches := fsolve(subs(val,U(x))=0,x,300..5
00);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 15 "Convert to feet" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "ans3feet := ans3inches/12;" }}}}
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