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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "# Solution of Cauchy-Euler
 Equations\n\n# filename = 04.6.mws\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "restart;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
89 "# Example 1\ny:='y':\ngensol:=dsolve(3*x^2*diff(y(x),x$2)-\n2*x*di
ff(y(x),x)+2*y(x)=0,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
109 "cvals:=seq(-2+2*i,i=0..2):\ntoplot:=\{seq(seq(rhs(gensol),_C1=cva
ls),_C2=cvals)\}:\nplot(toplot,x=0..12,y=-6..6);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 9 "Example 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 71 "y:='y':\ngensol:=dsolve(x^2*diff(y(x),x$2)-\nx*diff(y(x),x)+y(x)
=0,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "toplot:=\{seq(
seq(rhs(gensol),_C1=-1..1),_C2=-1..1)\}:\nplot(toplot,x=0..10,y=-5..5)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "y:='y':\ngensol:=dsolv
e(\nx^2*diff(y(x),x$2)-5*x*diff(y(x),x)+10*y(x)=0,y(x));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "assign(gensol);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 5 "y(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 11 "eval(y(1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "cval
s:=eval(solve(\{subs(x=1,y(x))=a,subs(x=1,diff(y(x),x))=b\},\{_C1,_C2
\}));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "sol:=subs(cvals,y(
x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "vals:=seq(-2+2*i,i=
0..2):\ntoplot:=\{seq(seq(sol,a=vals),b=vals)\}:\nplot(toplot,x=0..30)
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Example 4" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 81 "x:='x':y:='y':\neq:=2*x^3*diff(y(x),x$3)-4*
x^2*diff(y(x),x$2)-20*x*diff(y(x),x)=0:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "y:=x->x^m:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
3 "eq;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "factor(eq);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "mvals:=solve(factor(eq),m);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "y:='y':\ngensol:=dsolve
(eq,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "toplot:=\{seq
(seq(subs(_C1=0,rhs(gensol)),_C2=-1..1),_C3=-1..1)\}:\nplot(toplot,x=0
..2,y=-10..10);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Example 5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "y:='y':\neq:=x^4*diff(y(x),
x$4)+4*x^3*diff(y(x),x$3)+\n11*x^2*diff(y(x),x$2)-9*x*diff(y(x),x)+\n9
*y(x)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "step_1:=factor(
eval(subs(y(x)=x^m,eq)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "step_2:=solve(step_1,m);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 63 "y:=x->c[1]*cos(3*ln(x))+c[2]*sin(3*ln(x))+\nc[3]*x+c[4]*x*ln(x):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "d1:=simplify(diff(y(x),
x));\nd2:=simplify(diff(y(x),x$2));\nd3:=simplify(diff(y(x),x$3));" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "cvals:=solve(\{y(1)=1,D(y)(
1)=-9,(D@@2)(y)(1)=27,(D@@3)(y)(1)=1\});" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "sol:=subs(cvals,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "plot(sol,x=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "plot(sol,x=0..0.1);\nplot(sol,x=0..0.001);\nplot(sol,
x=0..0.00001);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "y:='y':\n
dsolve(\{eq,y(1)=1,D(y)(1)=-9,(D@@2)(y)(1)=27,(D@@3)(y)(1)=1\},y(x));
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Example 6" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 80 "x:='x':y:='y':\ngensol:=dsolve(x^2*diff(y(x)
,x$2)-\nx*diff(y(x),x)+5*y(x)=x,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "homsol:=dsolve(x^2*diff(y(x),x$2)-\nx*diff(y(x),x)+5*
y(x)=0,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "S:=eulerso
ls(x^2*diff(y(x),x$2)-\nx*diff(y(x),x)+5*y(x)=0,y(x));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "with(linalg):\nws:=simplify(det(wro
nskian(S,x)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=x->1/x
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "u1prime:=-S[2]*f(x)/ws
:\nu2prime:=S[1]*f(x)/ws:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
29 "u1:=x->int(u1prime,x):\nu1(x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "u2:=x->int(u2prime,x):\nu2(x);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 46 "yp:=x->S[1]*u1(x)+S[2]*u2(x):\nsimplify(yp(x))
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "yh:=x->c[1]*S[1]+c[2]*
S[2]:\nyh(x);\ny:=x->yh(x)+yp(x):\nsimplify(y(x));" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 68 "toplot:=\{seq(seq(y(x),c[1]=-1..1),c[2]=-1.
.1)\}:\nplot(toplot,x=0..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0
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