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{SECT 0 {EXCHG {PARA 210 "" 0 "" {TEXT 207 199 "Maple worksheet for Be
ssel's equation.  Here are some examples of ODEs that are Bessel equat
ions, or transformable into Bessel equations, so that their solutions \+
can be written as Bessel functions.  " }{TEXT 207 0 "" }}{PARA 210 "" 
0 "" {TEXT 207 58 "Created by A. Foster on Sept 2003, last modified Ma
y 2005." }{TEXT 207 0 "" }}{PARA 209 "> " 0 "" {MPLTEXT 1 0 57 "eqn:= \+
x^2*diff(y(x),x$2) +x*diff(y(x),x) +(x^2-9)*y(x)=0;" }{MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 207 37 "Get Maple to solve eqn by \+
any method:" }{TEXT 207 0 "" }}{PARA 209 "> " 0 "" {MPLTEXT 1 0 17 "ds
olve(eqn,y(x));" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 
207 452 "Maple gives the solution in terms of Bessel functions: y(x) =
 c1 J_3 (x) + c2 Y_3 (x).  That is, the general solution is given as a
n arbitrary constant (c1) times the Bessel function of the first kind \+
of order 3, plus another arbitrary constant (c2) times the Bessel func
tion of the second kind of order 3.  If you prefer, you can get Maple \+
to use J (instead of BesselJ) and Y (instead of BesselY) for the Besse
l functions of the first and second kind:" }{TEXT 207 0 "" }}}{EXCHG 
{PARA 209 "> " 0 "" {MPLTEXT 1 0 27 "alias(J=BesselJ,Y=BesselY);" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 17 "dsolv
e(eqn,y(x));" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 
200 104 "Okay, let's solve a different equation.  It is the same as th
e first equation, except for a pesky \"5\":" }{TEXT 200 0 "" }}}
{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 60 "eqn2:= x^2*diff(y(x),x$2) \+
+5*x*diff(y(x),x) +(x^2-9)*y(x)=0;" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 209 "> " 0 "" {MPLTEXT 1 0 18 "dsolve(eqn2,y(x));" }{MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 207 490 "Maple gives the gene
ral solution as y(x) = c1 x^(-2) J_nu (x) + c2 x^(-2) Y_nu (x), where \+
nu = sqrt(13).  Since nu is not an integer here, it would be preferabl
e to write the solution entirely in terms of Bessel functions of the f
irst kind: y(x) = c1 x^(-2) J_nu (x) + c_2 x^(-2) J_(-nu) (x).  But Ma
ple likes to use a Bessel function of the second kind \"Y_nu\" even wh
en nu is not an integer, by using a conversion formula Y_nu (x) = ( J_
(nu) (x) cos(nu pi) -  J_(-nu) (x) )/ sin(nu pi). " }{TEXT 207 0 "" }}
}{EXCHG {PARA 206 "" 0 "" {TEXT 200 94 "Here is another ODE.  It is no
t so obvious that the solution will consist of Bessel functions:" }
{TEXT 200 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 54 "eqn3:= x
^2*diff(y(x),x$2) +x*diff(y(x),x) +x^3*y(x)=0;" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 18 "dsolve(eqn3,y(x));" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 207 139 "The solut
ion is written as Bessel's functions of order zero: y(x) = c1 J_0 (t) \+
+ c2 Y_0 (t), with a change of variable: t = (2/3) x^(2/3). " }{TEXT 
207 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 207 17 "END OF WORKSHEET." 
}{TEXT 207 0 "" }{TEXT 207 2 "\n" }{TEXT 207 73 "Worksheet created by \+
Dr. A. Foster, Dept. of Mathematics and Statistics, " }{TEXT 207 0 "" 
}{TEXT 207 2 "\n" }{TEXT 207 60 "Memorial University of Newfoundland, \+
St. John's, NF, Canada." }{TEXT 207 0 "" }}}{PARA 205 "" 0 "" {TEXT 
201 0 "" }}{PARA 211 "" 0 "" {TEXT 208 0 "" }}}
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