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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Another example: finding a
 new solution." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "dex:=x*di
ff(y(x),x,x)+(1-2*x)*diff(y(x),x)+(x-1)*y(x)=0;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 22 "subs(y(x)=exp(x),dex);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 12 "simplify(\");" }{TEXT -1 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 34 "So it is a solution.  Get another:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "subs(y(x)=u(x)*exp(x),dex);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "des:=simplify(\");" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dsolve(des,u(x));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Finding constants (to solve IVP):
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "First let's compute a Wronski
an. It is in the linear algebra package." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "with(linalg): Wron:=Wronskian([exp(t),t^2-2*t+2],t);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "det(Wron); solve(\"=0,t
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(\{a+2*b=1,a-2*b
=0\},\{a,b\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "4.5  #11  Gener
al solution." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "d11:=4*diff
(w(x),x,x)+20*diff(w(x),x)+25*w(x)=0;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "dsolve(d11,w(x));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "factor(4*r^2+20*r+25);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 12 "4.5  #19 IVP" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
67 "i19:=\{diff(y(x),x,x)-4*diff(y(x),x)-5*y(x)=0, y(-1)=3, D(y)(-1)=9
\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dsolve(i19,y(x));" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 49 "4.5  #21  First order constant coefficien
t.  (a) " }{XPPEDIT 18 0 "a*diff(y(x),x)+b*y(x)=0;" "/,&*&%\"aG\"\"\"-
%%diffG6$-%\"yG6#%\"xGF-F&F&*&%\"bGF&-F+6#F-F&F&\"\"!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "d21:=a*diff(y(x),x)+b*y(x) = 0;" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dsolve(d21,y(x));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "4.5  #46  A third order Cauchy-Eul
er equation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "d46:=a*x^3*
diff(y(x),x$3)+b*x^2*diff(y(x),x,x)+ c*x*diff(y(x),x)+d*y(x)=0;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "s46:=subs(x=exp(t),d46);" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "So it doesn't work and you have t
o do it by hand!  " }}{PARA 0 "" 0 "" {TEXT -1 60 "Just use the chain \+
rule (and use the chain rule, and ...).  " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 50 "E:=exp(1);subs(x*diff(y(x),x)=diff(y(E^t),t),d46);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "So it won't do that either!  \+
 Let's just do the whole mess by hand." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "4.5  #49." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
28 "p49:=3*r^3+18*r^2+13*r-19=0;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "r49s:=solve(p49,r);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "evalf(r49s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 24 "Digits:=30; evalf(r49s);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
163 "Clearly the roots are real, and the complex roots are a result of
 the numerical method.  The same thing happens solving nonlinear equat
ions as one of you observed." }}}}{MARK "12 0 0" 29 }{VIEWOPTS 1 1 0 
2 1 1805 }