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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "y(x):=ln(u/x);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>-%\"yG6#%\"xG-%#lnG6#*&,&\"\"\"F-*&%\"CGF--%$ex
pG6#,$*&\"\"#!\"\"F'F5F-F-F-F6F'F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "simplify(eq);   #just checking" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "Q3" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "cp:=L^3-L^2-8*L+12;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cpG,**$)%\"LG\"\"$\"\"\"F**$)F(\"\"
#F*!\"\"*&\"\")F*F(F*F.\"#7F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 19 "gcd(cp,diff(cp,L));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"LG\"
\"\"\"\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "quo(cp,(L
-2)^2,L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"LG\"\"\"\"\"$F%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "yp:=(A*x+B)*exp(-2*x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ypG*&,&*&%\"AG\"\"\"%\"xGF)F)%\"BGF
)F)-%$expG6#,$*&\"\"#F)F*F)!\"\"F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 74 "eq:=diff(y(x),x,x,x)-diff(y(x),x,x)-8*diff(y(x),x)+12
*y(x)-32*x*exp(-2*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG,,-%%di
ffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"$\"\"\"-F'6$F)-F.6$F,\"\"#!\"\"*&\"\")
F1-F'6$F)F,F1F7*&\"#7F1F)F1F1*(\"#KF1F,F1-%$expG6#,$*&F6F1F,F1F7F1F7" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "simplify(eval(eq,y(x)=yp)
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\")\"\"\"-%$expG6#,$*&\"\"
#F&%\"xGF&!\"\"F&,*%\"AGF&*(F,F&F0F&F-F&F&*&F,F&%\"BGF&F&*&\"\"%F&F-F&
F.F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "A:=2:  B:=-1:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "y(x):=C1*exp(2*x)+C2*x*exp(
2*x)+C3*exp(-3*x)+yp;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"yG6#%\"x
G,**&%#C1G\"\"\"-%$expG6#,$*&\"\"#F+F'F+F+F+F+*(%#C2GF+F'F+F,F+F+*&%#C
3GF+-F-6#,$*&\"\"$F+F'F+!\"\"F+F+*&,&*&F1F+F'F+F+F+F;F+-F-6#,$*&F1F+F'
F+F;F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "simplify(eval(e
q));  #just checking" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "Q4" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 184 "A:=matrix([[8,-2,10,-3,16,-3,15,24],[3,6,24,1,1,9,
-42,58],[-1,6,16,4,1,-4,15,20],\n[-7,-1,-17,8,13,3,7,-7],[5,0,10,11,59
,9,2,92],[26,-2,46,-19,6,-25,73,24],\n[27,-8,30,-23,5,-21,58,4]]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7)7*\"\")!\"#\"#5!\"
$\"#;F-\"#:\"#C7*\"\"$\"\"'F0\"\"\"F4\"\"*!#U\"#e7*!\"\"F3F.\"\"%F4!\"
%F/\"#?7*!\"(F9!#<F*\"#8F2\"\"(F>7*\"\"&\"\"!F,\"#6\"#fF5\"\"#\"##*7*
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"> " 0 "" {MPLTEXT 1 0 28 "with(linalg):  gaussjord(A);" }}{PARA 7 "" 
1 "" {TEXT -1 80 "Warning, the protected names norm and trace have bee
n redefined and unprotected\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'ma
trixG6#7)7*\"\"\"\"\"!\"\"#F)\"\"$F)F(\"\"'7*F)F(F+F)!\"#F)F.F+7*F)F)F
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "[6,3,0,4,0,2,0]+[-2,-3,1,0,0,0,0]*c
1+[-3,2,0,-4,1,0,0]*c2+[-1,2,0,-3,0,4,1]*c3;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,*7)\"\"'\"\"$\"\"!\"\"%F'\"\"#F'\"\"\"*&7)!\"#!\"$F*F'
F'F'F'F*%#c1GF*F**&7)F.F)F'!\"%F*F'F'F*%#c2GF*F**&7)!\"\"F)F'F.F'F(F*F
*%#c3GF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "evalm(submatr
ix(A,1..7,1..7)&*%);   #just checking" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%'vectorG6#7)\"#C\"#e\"#?!\"(\"##*F'\"\"%" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 2 "Q5" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 
"cp:=m*(m-1)*(m-2)+2*m*(m-1)-8*m+12;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#cpG,**(%\"mG\"\"\",&F'F(F(!\"\"F(,&F'F(\"\"#F*F(F(*(F,F(F'F(F)F(F
(*&\"\")F(F'F(F*\"#7F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "g
cd(cp,diff(cp,m));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"mG\"\"\"\"
\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "quo(cp,(m-2)^2,
m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"mG\"\"\"\"\"$F%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "yp:=A*x^2*ln(x)^2;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#ypG*(%\"AG\"\"\")%\"xG\"\"#F')-%#lnG6#F)F
*F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "eq:=x^3*diff(y(x),x,
x,x)+2*x^2*diff(y(x),x,x)-8*x*diff(y(x),x)+12*y(x)-10*x^2;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#eqG,,*&)%\"xG\"\"$\"\"\"-%%diffG6$-%\"yG6
#F(-%\"$G6$F(F)F*F**(\"\"#F*)F(F5F*-F,6$F.-F26$F(F5F*F**(\"\")F*F(F*-F
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" {MPLTEXT 1 0 27 "simplify(eval(eq,y(x)=yp));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*(\"#5\"\"\")%\"xG\"\"#F&,&%\"AGF&F&!\"\"F&F&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "A:=1:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 36 "y(x):=C1*x^2+C2*x^2*ln(x)+C3/x^3+yp;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>-%\"yG6#%\"xG,**&%#C1G\"\"\")F'\"\"#F+F+*(%
#C2GF+F,F+-%#lnGF&F+F+*&%#C3GF+F'!\"$F+*&F,F+)F0F-F+F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "simplify(eq);  #just checking" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 2 "Q6" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
102 "A:=matrix([[15,15,-13,-13,13],[-5,-3,5,5,-5],[28,20,-21,-18,28],
\n[-19,-10,14,11,-19],[-4,-5,4,4,-1]]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"AG-%'matrixG6#7'7'\"#:F*!#8F+\"#87'!\"&!\"$\"\"&F0F.7'\"#G\"
#?!#@!#=F27'!#>!#5\"#9\"#6F77'!\"%F.\"\"%F=!\"\"" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 30 "with(linalg):  gausselim(A-L);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7'!\"&,&\"\"$!\"\"%\"LGF+\"\"&F-F(
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "evalm(-1/2+[1,-1]*sqrt(1/4+6
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(L^3-2*L^2-5*L+5,L=-3..4
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$e*R7)>MF0$\"3-))z$z\"*4e]%F07$$\"3'pmmmV,&eNF0$\"3eS6*4F\"pUpF07$$\"3
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L6\"Q!Fa\\l-%%VIEWG6$;F(Fa[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 15 "gaussjord(A-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'matrixG6#7'7'\"\"\"\"\"!F)F(F)7'F)F(F)F)F)7'F)F)F(\"\"#F)7'F)F)F)F)
F(7'F)F)F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gaussjord
(A+3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7'\"\"\"\"\"!F
)F)F)7'F)F(F)F)F)7'F)F)F(F(F)7'F)F)F)F)F(7'F)F)F)F)F)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "y:=evalm(C1*[-1,0,-2,1,0]*exp(2*x)+
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%'vectorG6#7',$*&%#C1G\"\"\"-%$expG6#,$*&\"\"#F,%\"xGF,F,F,!\"\"\"\"!,
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "simplify(evalm( [seq(diff(y[
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20 "6#-%'vectorG6#7'\"\"!F'F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "A[5,5]:=-2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "gausselim(A-L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7
'7'!\"&,&\"\"$!\"\"%\"LGF+\"\"&F-F(7'\"\"!,&#\"\"(F-\"\"\"*(\"#>F3F-F+
F,F3F3F(,&\"\")F+F,F+F/7'F/F/,$*(F-F3,&\"#8F+*&\"\"%F3F,F3F3F3,&F2F3*&
F5F3F,F3F3F+F3*(F;F3,&F7F3F,F3F3F?F+,&\"\"#F3F,F+7'F/F/F/,(*&#F3F-F3*$
)F,FDF3F3F3*&FHF3F,F3F3#\"\"'F-F+,$*(F-F+,*\"$)>F3*&\"$<\"F3F,F3F+*&F5
F3FJF3F+*&\"#9F3)F,F*F3F3F3F;F+F+7'F/F/F/F/,$*&,*\"#7F3*&F7F3F,F3F+FIF
+*$FWF3F3F3F;F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "pol:=L
^3-L^2-8*L+12:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "gcd(pol,d
iff(pol,L));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"#!\"\"%\"LG\"\"
\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "quo(pol,(L-2)^2,L);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"LG\"\"\"\"\"$F%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gaussjord(A-2);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'matrixG6#7'7'\"\"\"\"\"!F)F(F(7'F)F(F)F)F)7'F)F)F(
\"\"#F)7'F)F)F)F)F)F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "y1
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "gaussjord(stackmatrix((A-2)^2,[-1,0
,-2,1,0],[-1,0,0,0,1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG
6#7)7'\"\"\"\"\"!F)F)!\"\"7'F)F(F)F)!#67'F)F)F(F)!\"%7'F)F)F)F(!\"*7'F
)F)F)F)F)F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "s:=[1,11,4
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0 22 "y3:=(qs*x+s)*exp(2*x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "gaussjord(A+3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7
'7'\"\"\"\"\"!F)F)\"\"$7'F)F(F)F)!\"\"7'F)F)F(F(\"\"#7'F)F)F)F)F)F/" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "y4:=[0,0,-1,1,0]*exp(-3*x)
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{MPLTEXT 1 0 40 "y:=evalm(C1*y1+C2*y2+C3*y3+C4*y4+C5*y5):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "simplify( evalm( [seq(diff(y[i],x),
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%'vectorG6#7'\"\"!F'F'F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "Q7" 
}{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "A:=matrix([[-4,2,-2],[
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" 0 "" {MPLTEXT 1 0 30 "with(linalg):  gausselim(A-L);" }}{PARA 7 "" 
1 "" {TEXT -1 80 "Warning, the protected names norm and trace have bee
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\"$!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gaussjord(A-3);
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(!\"\"7%F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "y1:=[0,1,
1]*exp(3*t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gaussjord(A
+2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"\"!\"\"\"
\"!7%F*F*F(7%F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "y2:=
[1,1,0]*exp(-2*t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "gauss
jord(stackmatrix((A+2)^2,[1,1,0]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%'matrixG6#7&7%\"\"\"\"\"!F(7%F)F(!\"\"7%F)F)F)F," }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 61 "s:=[-1,1,1]:  qs:=evalm((A+2)&*s);    y3:
=(qs*t+s)*exp(-2*t):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#qsG-%'vecto
rG6#7%\"\"#F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "a1:=[
2,-8,-8]: a0:=[0,0,1]: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "
b1:=linsolve(A,-a1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b1G-%'vecto
rG6#7%\"\"!F)\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "b0:=
linsolve(A,b1-a0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b0G-%'vectorG
6#7%\"\"!F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "yp:=b1*t+b
0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "y:=evalm(C1*y1+C2*y2+
C3*y3+yp):   #just checking" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
49 "simplify(evalm([seq(diff(y[i],t),i=1..3)]-A&*y));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'vectorG6#7%,$*&\"\"#\"\"\"%\"tGF*F*,$*&\"\")F*F
+F*!\"\",&F*F**&F.F*F+F*F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "Q8" 
}{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "y=C*x+1/(1+C);  #  reg
ular solutions" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,&*&%\"CG\"\"
\"%\"xGF(F(*&F(F(,&F(F(F'F(!\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "x=diff(-1/(1+p),p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/%\"xG*&\"\"\"F&*$),&F&F&%\"pGF&\"\"#F&!\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 15 "p:=1/sqrt(x)-1:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "expand(x*p+1/(1+p)); #first singular" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,&*&\"\"#\"\"\"%\"xG#F&F%F&F'!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "simplify(%-x*diff(%,x)-1/(1+diff(%,
x)));  #just checking" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "p:=-1/sqrt(x)-1:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "expand(x*p+1/(1+p));    #second sin
gular" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#\"\"\"%\"xG#F&F%!\"
\"F'F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "simplify(%-x*diff
(%,x)-1/(1+diff(%,x)));  #just checking" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "Q9" }{MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "eq:=(1-x^2)*diff(y(x),x,x)-4*x*diff
(y(x),x)-2*y(x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "simplif
y(eval(eq,y(x)=x^i));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&)%\"xG,&%
\"iG\"\"\"\"\"#!\"\"F))F(F*F)F)*&F%F)F(F)F+*&)F&F(F)F,F)F+*(\"\"$F)F/F
)F(F)F+*&F*F)F/F)F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "sum of   c
i*i*(i-1)x^(i-2)  -  ci*(i+1)*(i+2)x^i      implies      c[i+2] = c[i]
" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sum(
x^(2*k),k=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F
%,&*$)%\"xG\"\"#F%F%F%!\"\"F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 29 "sum(x^(2*k+1),k=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#,$*&%\"xG\"\"\",&*$)F%\"\"#F&F&F&!\"\"F+F+" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 65 "simplify(eval(eq,y(x)=C1/(1-x^2)+C2*x/(1-x^2)));
  # just checking" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 4 "Q10:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "expand(r*(r-1)-r+1)   #indicial equation;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"rG\"\"#\"\"\"F(*&F'F(F&F(!\"\"F(F(" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "r=1 is double root" }{MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "eq:=x^2*(1-x^2)*
diff(y(x),x,x)-(3*x^3+x)*diff(y(x),x)+(1-x^2)*y(x):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 32 "simplify(eval(eq,y(x)=x^(i+1)));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&)%\"xG,&%\"iG\"\"\"F)F)F))F(\"\"#F
)F)*&)F&,&\"\"$F)F(F)F)F*F)!\"\"*(\"\"%F)F-F)F(F)F0*&F2F)F-F)F0" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "sum   ci i^2 x^(i+1)  - ci (i+2)^
2 x^(i+3)        or    c1 x^2 + sum  c[i+2](i+2)^2 x^(i+3) - ci (i+2)^
2 x^(i+3)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 33 "y1:=x*sum(x^(2*k),k=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#y1G,$*&%\"xG\"\"\",&*$)F'\"\"#F(F(F(!\"\"F-F-" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "simplify(eval(eq,y(x)=y1));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "simplify(eval(eq,y(x)=c(x)*y1));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&)%\"xG\"\"#\"\"\",&*&-%%diffG6$-%\"cG6#F%-%\"$G6$F%F&
F'F%F'F'-F+6$F-F%F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "in
t(1/z,z)=-int(1/x,x)+ln(C1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#ln
G6#%\"zG,&-F%6#%\"xG!\"\"-F%6#%#C1G\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 15 "int(C1/x,x)+C2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
,&*&%#C1G\"\"\"-%#lnG6#%\"xGF&F&%#C2GF&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "simplify(eval(eq,y(x)=%*y1));  # verifying" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 
"Q11" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "r
estart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "eq0:=diff(y(x),x
,x)-diff(y(x),x)*(1-diff(y(x),x)/y(x));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$eq0G,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#\"\"\"*&-F'6$F)F,
F1,&F1F1*&F3F1F)!\"\"F7F1F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
39 "eq1:=z(y)*diff(z(y),y)-z(y)*(1-z(y)/y);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$eq1G,&*&-%\"zG6#%\"yG\"\"\"-%%diffG6$F'F*F+F+*&F'F+,
&F+F+*&F'F+F*!\"\"F2F+F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 
"eq2:=diff(z(y),y)+z(y)/y-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2
G,(-%%diffG6$-%\"zG6#%\"yGF,\"\"\"*&F)F-F,!\"\"F-F-F/" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "int(1/z,z)=-int(1/y,y)+ln(c);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#%\"zG,&-F%6#%\"yG!\"\"-F%6#%
\"cG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "simplify(eval
(eq2,z(y)=c(y)/y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&-%%diffG6
$-%\"cG6#%\"yGF,!\"\"F,\"\"\"F.F,F-F-" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 14 "z = y/2 + C1/y" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "int(1/(y/2+C1/y),y)=x+ln(C2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%#lnG6#,&*$)%\"yG\"\"#\"\"\"F,*&F+F,%#C1GF,F,,&%
\"xGF,-F%6#%#C2GF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "ys:=-
sqrt(C2*exp(x)+C1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ysG,$*$,&*&%
#C2G\"\"\"-%$expG6#%\"xGF*F*%#C1GF*#F*\"\"#!\"\"" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 45 "simplify(eval(eq0,y(x)=ys));  # just checking
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 11 "diff(ys,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&
#\"\"\"\"\"#F&*(,&*&%#C2GF&-%$expG6#%\"xGF&F&%#C1GF&#!\"\"F'F+F&F,F&F&
F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "implies  C2 = 1 and  C1 = 0
  " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "ys
:=-exp(x/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ysG,$-%$expG6#,$*&
\"\"#!\"\"%\"xG\"\"\"F.F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
66 "simplify(eval(eq0,y(x)=ys));  eval(ys,x=0);  eval(diff(ys,x),x=0);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\"\"\"#" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "Q12" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "diff(y(x)-2/(C+x^2),x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%%diffG6$-%\"yG6#%\"xGF*\"\"\"*(\"
\"%F+,&%\"CGF+*$)F*\"\"#F+F+!\"#F*F+F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "simplify(eval(%,(C+x^2)=2/y(x)));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,&-%%diffG6$-%\"yG6#%\"xGF*\"\"\"*&)F'\"\"#F+F*F+F+" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eq:=diff(y(x),x)-1/x/y(x)
^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG,&-%%diffG6$-%\"yG6#%\"xG
F,\"\"\"*&F-F-*&F,F-)F)\"\"#F-!\"\"F2" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "int(y^2,y)=int(1/x,x)+C/3;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,$*&\"\"$!\"\"%\"yGF&\"\"\",&-%#lnG6#%\"xGF)*&F&F'%\"C
GF)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "y=(C+3*ln(x))^(1/3
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG*$),&%\"CG\"\"\"*&\"\"$F)-
%#lnG6#%\"xGF)F)#F)F+F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{MARK "144 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }