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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 82 " Transforming general linear second-order eigenval
ue problems into Sturm-Liouville" }}{PARA 0 "" 0 "" {TEXT -1 21 " eige
nvalue problems:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 10 "To convert" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 41 "a1(x) u'' + a2(x) u' + a3(x) u = lambda u" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "to SL-form:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "-(p(x) u'
)' + q(x) u = lambda w(x) u" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 34 "define a1-a3 below and hit return." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 32 "a1:=x->1-x^2:a2:=x->-x:a3:=x->0:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "DEQN:=a1(x)*diff(u(x),x$2)+a
2(x)*diff(u(x),x)+a3(x)*u(x)=lambda*u(x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%DEQNG/,&*&,&\"\"\"F)*$%\"xG\"\"#!\"\"F)-%%diffG6$-F/
6$-%\"uG6#F+F+F+F)F)*&F+F)F1F)F-*&%'lambdaGF)F3F)" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "     p,q and w are
 computed below:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 37 "p:=simplify(exp(int(a2(x)/a1(x),x)));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG*&,&!\"\"\"\"\"%\"xGF(#F(\"\"#,&
F(F(F)F(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "q:=-a3(x)/a1(
x)*p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG\"\"!" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 22 "w:=-simplify(p/a1(x));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"wG*(,&!\"\"\"\"\"%\"xGF(#F(\"\"#,&F(F(F)F(F*,&F'F
(*$F)F+F(F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "  The transformed \+
differential equation (SL form) is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "DEQNT:=-diff(p*diff(u(x),x
),x)+q*u(x)=omega*lambda*u(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&D
EQNTG/,(*(,&!\"\"\"\"\"%\"xGF*#F)\"\"#,&F*F*F+F*#F*F--%%diffG6$-%\"uG6
#F+F+F*F,*(F(F/F.F,F0F*F,*(F(F/F.F/-F16$F0F+F*F)*(%&omegaGF*%'lambdaGF
*F3F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 45 "     As a check the following should be zero." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "sim
plify(lhs(DEQNT/w)-lhs(DEQN));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"
!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "10 2 0" 0 
}{VIEWOPTS 1 1 0 1 1 1803 }