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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 14 "Sample test 2." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 12 "Question 1. " 
}{TEXT -1 114 "Integrate the following function. Do not use Maple's in
t, but use Maple's ratsols function in the DEtools package." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Notes:" }}{PARA 
0 "" 0 "" {TEXT -1 61 "1) See the worksheet on the web for week 8 on h
ow to do this." }}{PARA 0 "" 0 "" {TEXT -1 135 "2) During the actual t
est you can not use the worksheets on the web, you will have to be abl
e to do the following question on your own." }}{PARA 0 "" 0 "" {TEXT 
-1 500 "3) The worksheet on the web treats the more general case f in \+
K(theta) where K=C(x). It handles f by reducing it to a new f that is \+
in K[theta,theta^(-1)], so the new f is a Laurent polynomial in theta \+
(i.e. a polynomial where you may have negative powers of theta). Going
 from f in K(theta) to new f in K[theta,theta^(-1)] works in the same \+
way as in the logarithmic case, so that was already covered in test 1.
 So (and because of time constraints) we'll start with f already in K[
theta,theta^(-1)]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "f := \+
(x^2-1-x)/x^3*exp(x+1/x)+(2*x-x^2-1+2*x^4)/x^4/exp(x+1/x)+2/x*(-x+2*x^
2-2)/exp(x+1/x)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,(*&*&,(*$
)%\"xG\"\"#\"\"\"\"\"\"!\"\"F.F+F/F.-%$expG6#,&F+F.*&F-F-F+!\"\"F.F.F-
*$)F+\"\"$F-F5F.*&,*F+F,F)F/F/F.*$)F+\"\"%F-F,F-*&)F+\"\"%F-F0\"\"\"F5
F.*&,(F+F/F)F,!\"#F.F-*&F+\"\"\")F0\"\"#F-F5F," }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 258 11 "Question 2." }{TEXT -1 98 " Compute the indicial equatio
n and the exponents of L at the following two points: x=0  and  x=1.\n
" }}{PARA 0 "" 0 "" {TEXT -1 6 "Notes:" }}{PARA 0 "" 0 "" {TEXT -1 33 
"1) See the worksheets of week 12." }}{PARA 0 "" 0 "" {TEXT -1 145 "2)
 Maple's gen_exp command gives you the exponents but not the indicial \+
equation, so you still have to know how to compute the indicial equati
on." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "_Envdiffopdomain:=[D
x,x]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "L:=x^3*(x-1)^3*Dx^
3+(x^2-x)*Dx+2+24*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,**()%\"
xG\"\"$\"\"\"),&F(\"\"\"!\"\"F-F)F*)%#DxGF)F*F-*&,&*$)F(\"\"#F*F-F(F.F
-F0F-F-F5F-F(\"#C" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 12 "Question 3.
 " }{TEXT -1 336 "Consider the following differential operator L. Comp
ute the rational solutions of L. You may not use ratsols or dsolve. In
stead, use the exponents of L given below, to find a form for the rati
onal solutions with unknown coefficients, find equations for the unkno
wn coefficients, solve the equations and find all rational solutions o
f L." }}{PARA 0 "" 0 "" {TEXT -1 42 "Are all solutions of L rational f
unctions?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "L:=Dx^4-24*(2
3*x^3-27*x^2+13*x-2)/(x-1)^2/(4*x-1)/x^2*Dx^2-48*(26*x^3-9*x^2-4*x+1)/
(x-1)^2/(4*x-1)/x^3*Dx+240*(15*x^2-9*x+1)/(x-1)^2/(4*x-1)/x^3;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,**$)%#DxG\"\"%\"\"\"\"\"\"*&*&,
**$)%\"xG\"\"$F*\"#B*$)F1\"\"#F*!#FF1\"#8!\"#F+F+)F(F6F*F**(),&F1F+!\"
\"F+\"\"#F*,&F1F)F>F+\"\"\")F1\"\"#F*!\"\"!#C*&*&,*F/\"#EF4!\"*F1!\"%F
+F+F+F(F+F**()F=\"\"#F*F@\"\"\")F1\"\"$F*FD!#[*&,(F4\"#:F1FJF+F+F**()F
=\"\"#F*F@\"\"\")F1\"\"$F*FD\"$S#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
38 "You may use the following information:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 262 "finite_singularities:=[seq(i[1],i=factors(denom(L)
)[2])];\nsingularities:=[op(map(RootOf,finite_singularities)), infinit
y];\nfor i in singularities do\n  \"The smallest integer exponent at x
=\", i, \"is\",\n   gen_exp(L,T,x=i,'restrict_to'=\{'minimal',integer
\})[1][1]\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%5finite_singularit
iesG7%,&%\"xG\"\"\"#!\"\"\"\"%F(,&F'F(F*F(F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%.singularitiesG7&#\"\"\"\"\"%F'\"\"!%)infinityG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6&QDThe~smallest~integer~exponent~at~x=6
\"#\"\"\"\"\"%Q#isF$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&QDThe~sma
llest~integer~exponent~at~x=6\"\"\"\"Q#isF$!\"&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6&QDThe~smallest~integer~exponent~at~x=6\"\"\"!Q#isF$!\"&
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&QDThe~smallest~integer~exponent~at
~x=6\"%)infinityGQ#isF$!#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}}{MARK "10 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }