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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 41 "ORDINARY DIFFERENTIAL EQUATIONS POWERTOOL" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 33 "Unit 31 -- Power S
eries Solutions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{URLLINK 17 "Prof. Douglas B. Meade" 4 "http://www.math.sc.edu/~meade/
" "" }}{PARA 256 "" 0 "" {URLLINK 17 "Industrial Mathematics Institute
" 4 "http://www.math.sc.edu/~IMI/" "" }}{PARA 256 "" 0 "" {URLLINK 17 
"Department of Mathematics" 4 "http://www.math.sc.edu/" "" }}{PARA 
256 "" 0 "" {URLLINK 17 "University of South Carolina" 4 "http://www.s
c.edu/" "" }}{PARA 256 "" 0 "" {TEXT -1 19 "Columbia, SC 29208\n" }}
{PARA 256 "" 0 "" {TEXT -1 7 "URL:   " }{URLLINK 17 "http://www.math.s
c.edu/~meade/" 4 "http://www.math.sc.edu/~meade/" "" }}{PARA 256 "" 0 
"" {TEXT -1 25 "E-mail: meade@math.sc.edu" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 256 "" 0 "" {TEXT -1 38 "Copyright \251  2001  by Douglas
 B. Meade" }}{PARA 256 "" 0 "" {TEXT -1 19 "All rights reserved" }}
{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 67 "-----
--------------------------------------------------------------" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "Outline of Unit 31" }}
{EXCHG {PARA 14 "" 0 "" {HYPERLNK 17 "31.A" 1 "" "31.A" }{TEXT -1 39 "
 Taylor Series Methods for Solving IVPs" }}{PARA 14 "" 0 "" {HYPERLNK 
17 "31.B" 1 "" "31.B" }{TEXT -1 23 " Power Series Solutions" }}{PARA 
14 "" 0 "" {HYPERLNK 17 "31.C" 1 "" "31.C" }{TEXT -1 29 " Ordinary and
 Singular Points" }}{PARA 257 "" 0 "" {HYPERLNK 17 "31.C-1" 1 "" "31.C
-1" }{TEXT -1 35 " Example 1: Polynomial Coefficients" }}{PARA 257 "" 
0 "" {HYPERLNK 17 "31.C-2" 1 "" "31.C-2" }{TEXT -1 33 " Example 2: Rat
ional Coefficients" }}{PARA 257 "" 0 "" {HYPERLNK 17 "31.C-3" 1 "" "31
.C-3" }{TEXT -1 31 " Example 3: Legendre's Equation" }}{PARA 14 "" 0 "
" {HYPERLNK 17 "31.D" 1 "" "31.D" }{TEXT -1 20 " Method of Frobenius" 
}}{PARA 257 "" 0 "" {HYPERLNK 17 "31.D-1" 1 "" "31.D-1" }{TEXT -1 55 "
 Example 1: Two Real Roots with a Noninteger Difference" }}{PARA 14 "
" 0 "" {HYPERLNK 17 "31.E" 1 "" "31.E" }{TEXT -1 20 " Legendre's Equat
ion" }}{PARA 14 "" 0 "" {HYPERLNK 17 "31.F" 1 "" "31.F" }{TEXT -1 18 "
 Bessel's Equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "I
nitialization" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with( DEtools ):" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 14 "with( plots ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
15 "with( linalg ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "with( orthop
oly ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 
{PARA 3 "" 0 "31.A" {TEXT -1 43 "31.A Taylor Series Methods for Solvin
g IVPs" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Consider the initial val
ue problem" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 41 "ode1 := (x+y(x)+1) * diff( y(x), x ) = 1;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "ic1 := y(0) = 1;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 80 "If the solution to this IVP has a Taylor series expansion at th
e initial point, " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 142 
", then approximations to the solution can be obtained from the corres
ponding Taylor polynomials. For example, the quintic Taylor polynomial
 is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "N := 5:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Order :=
 N+1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "form1 := y(x) = convert( s
eries( y(x), x=0 ), polynom );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "To completely determin
e this polynomial, it is necessary to compute the value of " }
{XPPEDIT 18 0 "y(0)" "6#-%\"yG6#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 
0 "eval( diff( y(x), x ), x=0 )" "6#-%%evalG6$-%%diffG6$-%\"yG6#%\"xGF
,/F,\"\"!" }{TEXT -1 7 ", ..., " }{XPPEDIT 18 0 "eval( diff( y(x), x$5
), x=0 )" "6#-%%evalG6$-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"&/F,\"\"!
" }{TEXT -1 60 ". The first of these values is simply the initial cond
ition." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "Dy0[0] := ic1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "The second can be obta
ined by substituting the initial condition into the differential equat
ion:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "Dode1[0] := simplify( convert( ode1, D ) );" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 57 "Dy0[1] := isolate( subs( x=0, ic1, Dode1[
0] ), D(y)(0) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "To compute the value of " }
{XPPEDIT 18 0 "eval( diff( y(x),x$2), x=0 )" "6#-%%evalG6$-%%diffG6$-%
\"yG6#%\"xG-%\"$G6$F,\"\"#/F,\"\"!" }{TEXT -1 40 ", differentiate the \+
ODE with respect to " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 16 " and \+
substitute " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 42 " and t
he lower-order derivatives to obtain" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Dode1[1] := simplify( con
vert( diff( ode1, x ), D ) ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "Dy
0[2]   := isolate( subs( x=0, Dy0[i] $ i=0..1, Dode1[1] ), (D@@2)(y)(0
) );\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 119 "This process can be continued to determine the
 value of the remaining derivatives of the solution at the initial poi
nt." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "for n from 2 to N-1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 58 "  Dode1[n] := simplify( convert( diff( ode1, x$n ), D ) );" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "  Dy0[n+1] := isolate( subs( x=0, D
y0[i] $ i=0..n, Dode1[n] ), (D@@(n+1))(y)(0) );" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "end do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "unassign
( 'n' );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "print( Dy0[n] $ n=0..N \+
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "The approximate solution ob
tained by this method is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "sol1 := eval( form1, [ Dy0[i] $ i=0
..N ] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 59 "The same solution can be obtained directl
y from Maple when " }{HYPERLNK 17 "type=series" 2 "dsolve,series" "" }
{TEXT -1 38 " is included in the argument list for " }{HYPERLNK 17 "ds
olve" 2 "dsolve" "" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infolevel[dsolve] := 3:" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "sol1a := dsolve( \{ ode1, ic1 \}, \+
y(x), type=series );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infolevel[d
solve] := 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 243 "Note the presence of the \"order\" term \+
in solution. This indicates that the result is an approximation. To pl
ot this solution it is necessary to eliminate the order term; the simp
lest way to do this is to convert the expression to a polynomial:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 35 "sol1b := convert( sol1a, polynom );" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "The degree \+
of Maple's series solution is determined by the value of the global va
riable " }{HYPERLNK 17 "Order" 2 "Order" "" }{TEXT -1 96 ". Thus, the \+
ninth-degree Taylor polynomial approximate solution to this IVP can be
 obtained with" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 12 "Order := 10:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 63 "convert( dsolve( \{ ode1, ic1 \}, y(x), type=series ), polynom )
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 270 "Note that this method extends to IVPs for higher-or
der differential equations provided it is possible to express the high
est order derivative as a differentiable function of the lower order d
erivatives of the unknown function (and the independent variable). For
 example," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 73 "ode2 := ( x^2 + 1 ) * diff( y(x), x$2 ) + x * diff(
 y(x), x ) + y(x) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "O
rder := 12:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "ic2 := y(0)=
1, D(y)(0)=-1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "sol2 := c
onvert( dsolve( \{ ode2, ic2 \}, y(x), type=series ), polynom );" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Order := 6:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 246 "The c
omputation of Taylor series expansions is typically limited by the abi
lity to compute and manipulate the derivatives of the ODE. A tool such
 as Maple is very helpful in this regard, but other methods can be mor
e effective in many situations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "31.B" {TEXT -1 27 "31.B Power Serie
s Solutions" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 325 "Another problem wi
th the Taylor polynomials is that they are approximate solutions. The \+
exact solution can be obtained only when a general pattern for the coe
fficients in the expansion can be determined. This general pattern to \+
the coefficients can often be found directly by substituting a general
 power series into the ODE." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 84 "Consider the problem of finding the general sol
ution to the first-order (linear) ODE" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "ode3 := diff( y(x), x )
 - 2 * y(x) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "The general form of a power series
 solution, based at the origin, is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "form3 := y(x) = Sum( a(k)*x
^k, k=0..infinity );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "When this solution is substituted
 into the ODE and simplified, the result is :" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "q1 := eval(
 ode3, form3 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "q2 := co
mbine( q1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 398 "While Maple is not quite up to manipulat
ing infinite power series to find the general relation between the coe
fficients, a little insight and skill can be used to obtain this infor
mation. Because this equation is first order and, as seen above, only \+
consecutive pairs of coefficients are related, it suffices to insert t
he general three-term sum into the ODE and manipulate the expression t
o obtain" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "tform3 := y(x) = Sum( a(n)*x^n, n=k-1..k+1 );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "q1 := eval( ode3, tform3 );
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "q2 := simplify( combine
( value( q1 ), power ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "The recurrence equation betw
een the coefficients can now be found to be" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "recur_eqn3 := map
( coeff, q2, x^k );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "The (general) solution to this rec
urrence equation can be found with Maple's " }{HYPERLNK 17 "rsolve" 2 
"rsolve" "" }{TEXT -1 8 " command" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "recur_sol3 := rsolve( recur_
eqn3, \{a\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "The general power series solution \+
to the ODE is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 34 "sol3 := subs( recur_sol3, form3 );" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 15 "To express the " }{XPPEDIT 18 0 "Gamma" "6#%&GammaG" }{TEXT -1 
11 " function, " }{HYPERLNK 17 "GAMMA" 2 "GAMMA" "" }{TEXT -1 14 ", in
 terms of " }{HYPERLNK 17 "factorials" 2 "factorial" "" }{TEXT -1 1 ",
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "convert( sol3, factorial );" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "Observe
 that attempts to verify that these functions are solutions to the ODE
 via odetest requires a little additional work :" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "q3 := odete
st( sol3, ode3 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "value(
 q3 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 106 "Because of the simplicity of this example, it \+
is possible to identify the function with this power series:" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "v
alue( sol3 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The constant " }{XPPEDIT 18 0 "a(0
)" "6#-%\"aG6#\"\"!" }{TEXT -1 86 " takes the place of the integration
 constant in the other solution methods. Note that " }{XPPEDIT 18 0 "y
(0)=a(0)" "6#/-%\"yG6#\"\"!-%\"aG6#F'" }{TEXT -1 95 ". With this insig
ht, this result is seen to be equivalent to the explicit solution foun
d using " }{HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT -1 1 ":" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 21 "dsolve( ode3, y(x) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "There are many examples fo
r which " }{HYPERLNK 17 "rsolve" 2 "rsolve" "" }{TEXT -1 177 " will no
t be able to solve the recurrence relation. In that case, it will be n
ecessary to construct a finite set of explicit equations and solve for
 the first few coefficients :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "recur_sol3t := solve( \{ rec
ur_eqn3 $ k=0..5 \}, \{ a(k) $ k=1..6 \} );" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 68 "collect( eval( value(subs(infinity=6,form3)), recu
r_sol3t ), a(0) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 0 {PARA 3 "" 0 "31.C" {TEXT -1 33 "31.C Ordinary and Singular Po
ints" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Consider the homogeneous s
econd-order linear ODE in normalized form" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "ode4 := dif
f( y(x), x$2 ) + PP(x) * diff( y(x), x ) + Q(x) * y(x) = 0;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 15 "(Note that the " }{HYPERLNK 17 "orthopoly" 2 "orthopoly" 
"" }{TEXT -1 23 " package uses the name " }{HYPERLNK 17 "P" 2 "orthopo
ly,P" "" }{TEXT -1 57 " for the Legendre polynomials; hence the use of
 the name " }{TEXT 19 2 "PP" }{TEXT -1 33 " for the coefficient in the
 ODE.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 
"If initial conditions are given at " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6
#\"\"!" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ic4 := y(x0) = y0, D(y)(x0) = y1;" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 58 "then a power series solution should be based at the poi
nt " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 2 " :" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "f
orm4 := Sum( a(k) * ( x - x0 )^k, k=0..infinity );" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "The
 point " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 57 " is cal
led an ordinary point of the ODE if the functions " }{XPPEDIT 18 0 "P(
x)" "6#-%\"PG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Q(x)" "6#-%
\"QG6#%\"xG" }{TEXT -1 43 " can be expressed as power series based at \+
" }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 2 ", " }{TEXT 256 
4 "i.e." }{TEXT -1 2 ", " }{XPPEDIT 18 0 "P(x)" "6#-%\"PG6#%\"xG" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "Q(x)" "6#-%\"QG6#%\"xG" }{TEXT -1 
22 " are real analytic at " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }
{TEXT -1 21 ". If at least one of " }{XPPEDIT 18 0 "P(x)" "6#-%\"PG6#%
\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Q(x)" "6#-%\"QG6#%\"xG" }
{TEXT -1 25 " is not real analytic at " }{XPPEDIT 18 0 "x[0]" "6#&%\"x
G6#\"\"!" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!
" }{TEXT -1 95 " is called a singular point of the ODE. A singular poi
nt is called a regular singular point if " }{XPPEDIT 18 0 "(x-x0)*P(x)
" "6#*&,&%\"xG\"\"\"%#x0G!\"\"F&-%\"PG6#F%F&" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "(x-x0)^2*Q(x)" "6#*&,&%\"xG\"\"\"%#x0G!\"\"\"\"#-%\"QG6
#F%F&" }{TEXT -1 22 " are real analytic at " }{XPPEDIT 18 0 "x[0]" "6#
&%\"xG6#\"\"!" }{TEXT -1 12 ", otherwise " }{XPPEDIT 18 0 "x[0]" "6#&%
\"xG6#\"\"!" }{TEXT -1 32 " is an irregular singular point." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "31
.C-1" {TEXT -1 41 "31.C-1 Example 1: Polynomial Coefficients" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "When the coefficient functions are
 polynomials, " }{TEXT 257 4 "e.g." }{TEXT -1 1 "," }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "coeff1 := \+
\{ PP(x) = 1 - x^3, Q(x) = x^2 \};" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "eval( ode4, coeff1 );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "all points " 
}{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 21 " are ordinary po
ints." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 
{PARA 4 "" 0 "31.C-2" {TEXT -1 39 "31.C-2 Example 2: Rational Coeffici
ents" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "When the coefficient funct
ions are rational functions, " }{TEXT 258 4 "e.g." }{TEXT -1 1 "," }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 55 "coeff2 := \{ PP(x) = 2/(1 - x)^2, Q(x) = x^(-2)/(x-2) \};" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eval( ode4, coeff2 );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 23 "all discontinuities of " }{XPPEDIT 18 0 "P(x)" "6#-%\"PG6
#%\"xG" }{TEXT -1 7 " or of " }{XPPEDIT 18 0 "Q(x)" "6#-%\"QG6#%\"xG" 
}{TEXT -1 21 " are singular points:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "singpts := `union`(op( map
( c -> discont( rhs( c ), x ), coeff2 ) ));" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "To decide
 if the singular points are regular or irregular, the definition can b
e applied or Maple's " }{HYPERLNK 17 "regularsp" 2 "DEtools,regularsp
" "" }{TEXT -1 19 " command (from the " }{HYPERLNK 17 "DEtools" 2 "DEt
ools" "" }{TEXT -1 22 " package) can be used:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "regularsp( \+
eval( ode4, coeff2 ), x, y(x) );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "and so " }
{XPPEDIT 18 0 "x[0]=1" "6#/&%\"xG6#\"\"!\"\"\"" }{TEXT -1 37 " must be
 an irregular singular point." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "3
1.C-3" {TEXT -1 37 "31.C-3 Example 3: Legendre's Equation" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 29 "Legendre's equation of order " }{XPPEDIT 
18 0 "p" "6#%\"pG" }{TEXT -1 3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "ode5 := ( 1 - x^2 ) * diff
( y(x), x$2 ) - 2*x * diff( y(x), x ) + p*(p+1) * y(x) = 0;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 6 "where " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 73 " is a n
onnegative constant. Written in normalized form, the coefficients " }
{XPPEDIT 18 0 "P(x)" "6#-%\"PG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 
18 0 "Q(x)" "6#-%\"QG6#%\"xG" }{TEXT -1 4 " are" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "q4 := eval(
 lhs(ode5), [ diff( y(x), x$2 ) = d2y, diff( y(x), x ) = dy, y(x)=y ] \+
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "coeff3 := \{ PP(x) = \+
coeff( q4, dy )/coeff( q4, d2y ), Q(x) = coeff( q4, y )/coeff( q4, d2y
 ) \};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 23 "The singular points are" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "singpts := \+
`union`(op( map( c -> discont( rhs( c ), x ), coeff3 ) ));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 52 "and both of these points are regular singular points" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "r
egularsp( ode5, x, y(x) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}}}{SECT 0 {PARA 3 "" 0 "31.D" {TEXT -1 24 "31.D Method of Frob
enius" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "The method for finding a \+
power series solution presented in " }{HYPERLNK 17 "Unit 31, Section B
" 1 "unit31.mws" "31.B" }{TEXT -1 276 " can be applied at any ordinary
 point of an ODE. The Method of Frobenius is guaranteed to find at lea
st one nontrivial solution in a neighborhood of a regular singular poi
nt. When the Method of Frobenius does not produce a second solution, r
eduction of order can be used (see " }{HYPERLNK 17 "Unit 19" 1 "unit19
.mws" "" }{TEXT -1 117 "). For an irregular singular point, the Method
 of Frobenius can be tried but may not produce a solution in all cases
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The \+
Method of Frobenius produces solutions of the form" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "form5 := y(
x) = Sum( a(k) * (x-x0)^(k+r), k=0..infinity );" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "When " 
}{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 73 " is a nonnegative integer, \+
the Frobenius solution is a power series, but " }{XPPEDIT 18 0 "r" "6#
%\"rG" }{TEXT -1 213 " does not have to be an integer -- it can be a n
egative or complex number. When a Frobenius solution is inserted into \+
the ODE, the condition for the vanishing of the lowest order term is a
 quadratic polynomial in " }{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 46 "
. The general form of the indicial equation is" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "r^2 + (p[0]-1)*r + q[0] = 0" "6#/,(*$%
\"rG\"\"#\"\"\"*&,&&%\"pG6#\"\"!F(F(!\"\"F(F&F(F(&%\"qG6#F.F(F." }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "p
[0]" "6#&%\"pG6#\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "q[0]" "6#&%
\"qG6#\"\"!" }{TEXT -1 58 " are the constant terms in the power series
 expansions of " }{XPPEDIT 18 0 "p(x) = x*P(x)" "6#/-%\"pG6#%\"xG*&F'
\"\"\"-%\"PG6#F'F)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "q(x) = x^2*Q(x
)" "6#/-%\"qG6#%\"xG*&F'\"\"#-%\"QG6#F'\"\"\"" }{TEXT -1 15 ", respect
ively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
158 "The full Method of Frobenius is long and exceedingly complicated.
 In the remainder of this worksheet the basic ideas will be explored i
n a number of examples." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "31.D-1" 
{TEXT -1 61 "31.D-1 Example 1: Two Real Roots with a Noninteger Differ
ence" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Consider the ODE" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "o
de6 := 2*x*(1+x) * diff( y(x), x$2 ) + (3+x) * diff( y(x), x ) - x * y
(x) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 48 "The regular singular points for this prob
lem are" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "regularsp( ode6, x, y(x) );" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "The indi
cial equation at the regular singular point " }{XPPEDIT 18 0 "x[0]=0" 
"6#/&%\"xG6#\"\"!F'" }{TEXT -1 3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ie1 := indicialeq( ode6
, x, 0, y(x) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "The indicial roots are" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ir1
 := solve( ie1, x );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "Because these roots do not diffe
r by an integer, each indicial root should lead to a nontrivial Froben
ius solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 53 "formF := y(x) = Sum( c(k) * x^(k+r), k=0..infini
ty ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "formF1 := eval( formF, [ c
=a, r=max(ir1) ] );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "formF2 := ev
al( formF, [ c=b, r=min(ir1) ] );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "To obtain th
e recurrence relations for the coefficients in the two solutions, work
 with truncated versions of the solution" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "tformF1 := eval( y(x
) = Sum( a(m) * x^(m+r), m=k-2..k+2 ), r=max(ir1) );" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 72 "tformF2 := eval( y(x) = Sum( b(m) * x^(m+r), m=k
-2..k+2 ), r=min(ir1) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 
"q51 := simplify( combine( value( eval( ode6, tformF1 ) ), power ) );
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "q52 := simplify( combine( value
( eval( ode6, tformF2 ) ), power ) );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "reqn1 := collect( map( coeff, q51, eval( x^(k+r), r=m
ax(ir1) ) )," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "                  [
 a(k+i) $ i=-3..3 ] );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "reqn2 := \+
collect( map( coeff, q52, eval( x^(k+r), r=min(ir1) ) )," }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 41 "                  [ b(k+i) $ i=-3..3 ] );" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 69 "While these recurrence relations cannot be solved explici
tly by Maple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 21 "rsolve( reqn1, \{a\} );" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "rsolve( reqn2, \{b\} );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "The first few
 terms are easily computed" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "q61 := isolate( reqn1, a(k+1) );" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "q62 := isolate( reqn2, b(k+1) );" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "unassign( 'a', 'b' );" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for k from 1 to 8 do" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 24 "  assign( \{ q61, q62\} );" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 48 "  print( 'a'(k+1) = a(k+1), 'b'(k+1) = b(k+1) );" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "unassign( 'k' );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "and so the (a
pproximate) solutions to the ODE are" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "q71 := value( subs( infin
ity=9, formF1 ) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "q72 := value(
 subs( infinity=9, formF2 ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "sol61 := sort( c
ollect( q71, [ a(0), a(1) ] ) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 
"sol62 := sort( collect( q72, [ b(0), b(1) ] ) );" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "To c
onclude, compare this solution to the series solution obtained by " }
{HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT -1 6 " with " }{HYPERLNK 
17 "type=series" 2 "dsolve,series" "" }{TEXT -1 2 " :" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Order :=
 10:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "sol6 := dsolve( ode6, y(x),
 type=series );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Order := 6:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 161 "To see the equivalence of these solutions,  observe that
 the sum of the two linearly independent solutions obtained from the M
ethod of Frobenius agrees with the " }{HYPERLNK 17 "dsolve" 2 "dsolve
" "" }{TEXT -1 35 " solution for a specific choice of " }{XPPEDIT 18 
0 "a(0)" "6#-%\"aG6#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a(1)" "6#-
%\"aG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b(0)" "6#-%\"bG6#\"\"!
" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "b(1)" "6#-%\"bG6#\"\"\"" }
{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 62 "q8 := rhs(sol61) + rhs(sol62) = convert( rhs(s
ol6), polynom );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "solve( \+
identity(q8,x), \{a(0),a(1),b(0),b(1)\} );" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA 3 "" 0 "31.E" {TEXT -1 24 "31.E
 Legendre's Equation" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Recall, fr
om " }{HYPERLNK 17 "Section 31.C-3" 1 "" "31.C-3" }{TEXT -1 36 ", that
 Legendre's equation of order " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT 
-1 26 " (a nonnegative constant)," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "ode5;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "has tw
o regular singular points" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "regularsp( ode5, x, y(x) );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 8 "Because " }{XPPEDIT 18 0 "x[0]=0" "6#/&%\"xG6#\"\"!F'" }
{TEXT -1 118 ", is an ordinary point, there should be no trouble simpl
y finding the general solution in the form of a power series :" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 49 "form5 := y(x) = Sum( a(k) * x^k, k=0..infinity );" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 118 "To obtain the recurrence relation for the coefficients in this
 solution, work with a truncated version of the solution" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "tfor
m5 := y(x) = Sum( a(m) * x^m, m=k-2..k+2 );" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 66 "q9 := simplify( combine( value( eval( ode5, tform5
 ) ), power ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "reqn3 :
= collect( map( coeff, q9, x^k ), [ a(k+i) $ i=-3..3 ] );" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 67 "While this recurrence relation cannot be solved explicitly by M
aple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "rsolve( reqn3, \{a\} );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "The first few
 terms are easily computed" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "q10 := factor( isolate( reqn3, a(k+
2) ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "unassign( 'a' );" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 20 "for k from 0 to 6 do" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "  assign( \{ q10 \} );" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 29 "  print( 'a'(k+2) = a(k+2) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 7 "end do;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "unassign( 'k' );" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 49 "and so the (approximate) solutions to the ODE are" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 77 "q11 := sort( collect( value( subs( infinity=8, form5 ) ), [ a(0)
, a(1) ] ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "To confirm this computation, compa
re this solution to the series solution obtained by " }{HYPERLNK 17 "d
solve" 2 "dsolve" "" }{TEXT -1 6 " with " }{HYPERLNK 17 "type=series" 
2 "dsolve,series" "" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Order := 9:" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 92 "sol5 := collect( convert( dsolve( ode5, y(x),
 type=series ), polynom ), [ y(0), D(y)(0) ] );" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "Order := 6:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 161 "To see the equivalence \+
of these solutions,  observe that the sum of the two linearly independ
ent solutions obtained from the Method of Frobenius agrees with the " 
}{HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT -1 35 " solution for a spe
cific choice of " }{XPPEDIT 18 0 "a(0)" "6#-%\"aG6#\"\"!" }{TEXT -1 5 
" and " }{XPPEDIT 18 0 "a(1)" "6#-%\"aG6#\"\"\"" }{TEXT -1 2 " :" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 28 "q12 := rhs(q11) = rhs(sol5):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
38 "solve( identity(q12,x), \{a(0),a(1)\} );" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "While we
 have not proven it, the series associated with " }{XPPEDIT 18 0 "a(0)
" "6#-%\"aG6#\"\"!" }{TEXT -1 32 " and the series associated with " }
{XPPEDIT 18 0 "a(1)" "6#-%\"aG6#\"\"\"" }{TEXT -1 2 ", " }{TEXT 259 4 
"i.e." }{TEXT -1 19 ", the two functions" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "sol51 := y[1](x) = e
val( rhs(q11), [ a(0)=1, a(1)=0 ] );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 56 "sol52 := y[2](x) = eval( rhs(q11), [ a(0)=0, a(1)=1 ] );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 68 "form a fundamental set of solutions to Legendre's equatio
n of order " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Additional inspection of these sol
utions (or the recurrence relations) when " }{XPPEDIT 18 0 "p" "6#%\"p
G" }{TEXT -1 61 " is an integer reveals that either the series of even
 terms (" }{XPPEDIT 18 0 "y[1]" "6#&%\"yG6#\"\"\"" }{TEXT -1 30 ") or \+
the series of odd terms (" }{XPPEDIT 18 0 "y[2]" "6#&%\"yG6#\"\"#" }
{TEXT -1 112 ") terminates. That is, one fundamental solution is a pol
ynomial. This polynomial, normalized to have value 1 at " }{XPPEDIT 
18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 38 ", is the Legendre polynomial
 of order " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for k \+
from 0 to 5 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "  if type(k,even)
 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "    q := eval( rhs(sol51),
 p=k );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "  else" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 33 "    q := eval( rhs(sol52), p=k );" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 9 "  end if;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " \+
 P[k] = q / eval( q, x=1 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end d
o;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 82 "The Legendre polynomials form an orthogonal family
 of functions in the sense that " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 " \+
Int( P[m](x)*P[n](x), x=-1..1 ) = 0" "6#/-%$IntG6$*&-&%\"PG6#%\"mG6#%
\"xG\"\"\"-&F*6#%\"nG6#F.F//F.;,$F/!\"\"F/\"\"!" }{TEXT -1 32 "   for \+
all nonnegative integers " }{XPPEDIT 18 0 "m <> n" "6#0%\"mG%\"nG" }
{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 47 "Maple provides the Legendre polynomials in the " }
{HYPERLNK 17 "orthopoly" 2 "orthopoly" "" }{TEXT -1 10 " package :" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "for k from 0 to 5 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "  P( \+
k, x );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do;" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 16 "unassign( 'k' ):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "plot( [
 P(k,x) $ k=0..6 ], x=-1..1, legend=[ seq( \"P\"||k, k=0..6 ) ] );" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 161 "The orthogonality of the (first few) Legendre polynomial
s is confirmed by the diagonal structure in the matrix of inner produc
ts of pairs of Legendre polynomials" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "matrix( 9, 9, (m,n) -> int
( P(m,x)*P(n,x), x=-1..1 ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{HYPERLNK 17 "Lege
ndreP" 2 "LegendreP" "" }{TEXT -1 5 " and " }{HYPERLNK 17 "LegendreQ" 
2 "LegendreQ" "" }{TEXT -1 110 " commands provide high level access to
 the Legendre functions of the first and second kind, respectively. Th
e " }{HYPERLNK 17 "series" 2 "series" "" }{TEXT -1 88 " command can be
 used to obtain a series expansion for the appropriate Legendre functi
on." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "series( LegendreP(1/2,x), x=0 );" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "31.F" {TEXT -1 
22 "31.F Bessel's Equation" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Bess
el's equation of order " }{XPPEDIT 18 0 "nu" "6#%#nuG" }{TEXT -1 3 " i
s" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 82 "ode7 := x^2 * diff( y(x), x$2 ) + x * diff( y(x), x )
 + ( x^2 - nu^2 ) * y(x) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "n
u" "6#%#nuG" }{TEXT -1 111 " is a nonnegative constant. All points are
 ordinary points except for the regular singular point at the origin.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "regularsp( ode7, x, y(x) );" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "Thus, a
 series solution at the origin can be found with the Method of Frobeni
us. The indicial equation at the origin is" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ie2 := indicialeq
( ode7, x, 0, y(x) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "with indicial roots" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ir2 \+
:= solve( ie2, x );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Because " }{XPPEDIT 18 0 "x[0]=0" "
6#/&%\"xG6#\"\"!F'" }{TEXT -1 100 " is a regular singular point for Be
ssel's equation, solutions are sought via the Method of Frobenius" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 53 "formF := y(x) = Sum( c(k) * x^(k+r), k=0..infinity );" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 39 "formF3 := eval( formF, [ c=a, r=nu ] );" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "formF4 := eval( formF, [ c=b, r=-
nu ] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 122 "To obtain the recurrence relations for t
he coefficients in the two solutions, work with truncated versions of \+
the solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 66 "tformF3 := eval( y(x) = Sum( a(m) * x^(m+r), m=k
-2..k+2 ), r=nu );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "tformF4 := ev
al( y(x) = Sum( b(m) * x^(m+r), m=k-2..k+2 ), r=-nu );" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "q131 := simplify( combine( value( e
val( ode7, tformF3 ) ), power ) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
69 "q132 := simplify( combine( value( eval( ode7, tformF4 ) ), power )
 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "reqn3 := collect( ma
p( coeff, q131, eval( x^(k+r), r=nu ) ), [ a(k+i) $ i=-3..3 ] );" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "reqn4 := collect( map( coeff, q132,
 eval( x^(k+r), r=-nu ) ), [ b(k+i) $ i=-3..3 ] );" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Whi
le these recurrence relations cannot be solved explicitly by Maple" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 21 "rsolve( reqn3, \{a\} );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "rs
olve( reqn4, \{b\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "The first few terms are easily c
omputed" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "q141 := factor( isolate( reqn3, a(k) ) );" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 41 "q142 := factor( isolate( reqn4, b(k) ) );" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 21 "unassign( 'a', 'b' );" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "for k from 2 to 9 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 26 "  assign( \{ q141, q142\} );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
40 "  print( 'a'(k) = a(k), 'b'(k) = b(k) );" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "end do;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "unassign
( 'k' );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 49 "and so the (approximate) solutions to the
 ODE are" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "q151 := value( subs( infinity=9, formF3 ) );" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "q152 := value( subs( infinity=9, fo
rmF4 ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "sol71 := sort( collect( q151, [ a(0
), a(1) ] ) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "sol72 := sort( co
llect( q152, [ b(0), b(1) ] ) );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "To confirm th
ese computations, compare these solution to the series solution obtain
ed by " }{HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT -1 6 " with " }
{HYPERLNK 17 "type=series" 2 "dsolve,series" "" }{TEXT -1 2 " :" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 12 "Order := 10:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "sol7 := dsolv
e( ode7, y(x), type=series );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Or
der := 6:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 160 "To see the equivalence of these solution
s, observe that the sum of the two linearly independent solutions obta
ined from the Method of Frobenius agrees with the " }{HYPERLNK 17 "dso
lve" 2 "dsolve" "" }{TEXT -1 35 " solution for a specific choice of " 
}{XPPEDIT 18 0 "a(0)" "6#-%\"aG6#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 
0 "a(1)" "6#-%\"aG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b(0)" "6#
-%\"bG6#\"\"!" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "b(1)" "6#-%\"bG6#
\"\"\"" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "q16 := rhs(sol71) + rhs(sol72) = co
nvert( rhs(sol7), polynom ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "sol
ve( identity(q16,x), \{a(0),a(1),b(0),b(1)\} );" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Note t
hat these series solutions are not well defined when " }{XPPEDIT 18 0 
"2*nu" "6#*&\"\"#\"\"\"%#nuGF%" }{TEXT -1 21 " is an integer. When " }
{XPPEDIT 18 0 "2*nu" "6#*&\"\"#\"\"\"%#nuGF%" }{TEXT -1 89 " is not an
 integer, a fundamental set of solutions to Bessel's equation of order
 nu is \{ " }{XPPEDIT 18 0 "J[nu](x)" "6#-&%\"JG6#%#nuG6#%\"xG" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "J[-nu](x)" "6#-&%\"JG6#,$%#nuG!\"\"6#%
\"xG" }{TEXT -1 2 " \}" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 58 "Define the Bessel's function (of the firs
t kind) of order " }{XPPEDIT 18 0 "nu" "6#%#nuG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "-nu" "6#,$%#nuG!\"\"" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "
nu" "6#%#nuG" }{TEXT -1 19 " nonnegative) to be" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "J[nu](x) = \+
eval( rhs(sol71), [ a(0)=1, a(1)=0 ] );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "J[-nu](x) = eval( rhs(sol72), [ b(0)=1, b(1)=0 ] );" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 35 "While it is tempting to state that " }{XPPEDIT 18 0 "\{
 J[nu](x), J[-nu](x) \}" "6#<$-&%\"JG6#%#nuG6#%\"xG-&F&6#,$F(!\"\"6#F*
" }{TEXT -1 65 " is a fundamental set of solutions to Bessel's equatio
n of order " }{XPPEDIT 18 0 "nu" "6#%#nuG" }{TEXT -1 12 ", note that \+
" }{XPPEDIT 18 0 "J[-0](x) = J[0](x)" "6#/-&%\"JG6#,$\"\"!!\"\"6#%\"xG
-&F&6#F)6#F," }{TEXT -1 10 " and that " }{XPPEDIT 18 0 "J[-nu]" "6#&%
\"JG6#,$%#nuG!\"\"" }{TEXT -1 26 " is not well-defined when " }
{XPPEDIT 18 0 "2*nu" "6#*&\"\"#\"\"\"%#nuGF%" }{TEXT -1 33 " is an eve
n positive integer. If " }{XPPEDIT 18 0 "nu" "6#%#nuG" }{TEXT -1 33 " \+
is not a positive integer, then " }{XPPEDIT 18 0 "\{J[nu](x), J[-nu](x
) \}" "6#<$-&%\"JG6#%#nuG6#%\"xG-&F&6#,$F(!\"\"6#F*" }{TEXT -1 65 " is
 a fundamental set of solutions to Bessel's equation of order " }
{XPPEDIT 18 0 "nu" "6#%#nuG" }{TEXT -1 5 ". If " }{XPPEDIT 18 0 "nu" "
6#%#nuG" }{TEXT -1 58 " is a positive integer, a fundamental set of so
lutions is " }{XPPEDIT 18 0 "\{ J[nu](x), Y[nu](x) \}" "6#<$-&%\"JG6#%
#nuG6#%\"xG-&%\"YG6#F(6#F*" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "Y[nu
](x)" "6#-&%\"YG6#%#nuG6#%\"xG" }{TEXT -1 50 ", the Bessel function of
 the second kind of order " }{XPPEDIT 18 0 "nu" "6#%#nuG" }{TEXT -1 
35 ", is defined for noninteger orders " }{XPPEDIT 18 0 "nu" "6#%#nuG
" }{TEXT -1 66 " to be a linear combination of the linearly independen
t functions " }{XPPEDIT 18 0 "J[nu](x)" "6#-&%\"JG6#%#nuG6#%\"xG" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "J[-nu](x)" "6#-&%\"JG6#,$%#nuG!\"\"
6#%\"xG" }{TEXT -1 2 " :" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Y[nu](x) = ( cos( nu*Pi ) * J[nu](x) - J[-nu](x) ) / si
n( nu*Pi )" "6#/-&%\"YG6#%#nuG6#%\"xG*&,&*&-%$cosG6#*&F(\"\"\"%#PiGF2F
2-&%\"JG6#F(6#F*F2F2-&F66#,$F(!\"\"6#F*F=F2-%$sinG6#*&F(F2F3F2F=" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "and, for \+
integer orders (" }{XPPEDIT 18 0 "nu=m" "6#/%#nuG%\"mG" }{TEXT -1 3 ")
 :" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Y[m](x) = Limit
( Y[nu](x), nu=m )" "6#/-&%\"YG6#%\"mG6#%\"xG-%&LimitG6$-&F&6#%#nuG6#F
*/F1F(" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 71 "Thus, the general solution to Bessel's equation of (
nonnegative) order " }{XPPEDIT 18 0 "nu" "6#%#nuG" }{TEXT -1 18 " can \+
be written as" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(x)
 = c[1] * J[nu](x) + c[2] * Y[nu](x)" "6#/-%\"yG6#%\"xG,&*&&%\"cG6#\"
\"\"F--&%\"JG6#%#nuG6#F'F-F-*&&F+6#\"\"#F--&%\"YG6#F26#F'F-F-" }{TEXT 
-1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
92 "Direct access to the Bessel functions of the first and second kind
 is provided with Maple's " }{HYPERLNK 17 "BesselJ" 2 "BesselJ" "" }
{TEXT -1 5 " and " }{HYPERLNK 17 "BesselY" 2 "BesselY" "" }{TEXT -1 
10 " commands." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 54 "Two important properties of Bessel functions are that " }
{XPPEDIT 18 0 "limit( J[nu](x), x=0, right ) = 0" "6#/-%&limitG6%-&%\"
JG6#%#nuG6#%\"xG/F-\"\"!%&rightGF/" }{TEXT -1 6 " and  " }{XPPEDIT 18 
0 "limit( Y[nu](x), x=0, right ) = -infinity" "6#/-%&limitG6%-&%\"YG6#
%#nuG6#%\"xG/F-\"\"!%&rightG,$%)infinityG!\"\"" }{TEXT -1 57 ". This p
roperty is essentially a result of the fact that " }{XPPEDIT 18 0 "J[n
u](x)" "6#-&%\"JG6#%#nuG6#%\"xG" }{TEXT -1 36 " is obtained from the i
ndicial root " }{XPPEDIT 18 0 "r=nu" "6#/%\"rG%#nuG" }{TEXT -1 9 " > 0
 and " }{XPPEDIT 18 0 "Y[nu](x)" "6#-&%\"YG6#%#nuG6#%\"xG" }{TEXT -1 
24 " from the indicial root " }{XPPEDIT 18 0 "r=-nu" "6#/%\"rG,$%#nuG!
\"\"" }{TEXT -1 63 " < 0. These properties can be observed in the foll
owing plots :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 55 "plot( [ BesselJ(1,x), BesselY(1,x) ], x=0..10, y
=-2..1," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "      legend=[ \"J[1]\",
 \"Y[1]\" ] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plot( [ B
esselJ(7/3,x), BesselY(7/3,x) ], x=0..10, y=-2..1," }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 38 "      legend=[ \"J[7/3]\", \"Y[7/3]\" ] );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 122 "As a final comment, the Bessel functions of half integer
 orders can be expressed in unexpectedly simple forms. For example" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 26 "for k from -5 to 5 by 2 do" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "  J[k/2](x) = BesselJ( k/2, x );" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 7 "end do;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "[Back to " }{HYPERLNK 17 "OD
E Powertool Table of Contents" 1 "unit00.mws" "" }{TEXT -1 1 "]" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }