{VERSION 3 0 "IBM INTEL NT" "3.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 
0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 256 "" 0 24 255 0 255 1 1 0 0 0 0 2 0 0 0 }{CSTYLE "" -1 257 "" 
0 24 255 0 255 1 1 0 0 0 0 2 0 0 0 }{CSTYLE "" -1 258 "" 0 24 255 0 
255 1 1 0 0 0 0 2 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }
{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 
1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading
 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 
8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 
0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE 
"Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 
0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 
-1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }
{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 
0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE 
"" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 
-1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" 
-1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 
-1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {PARA 18 "" 0 "" {TEXT -1 29 "Linear Differential Equations" }
}{PARA 18 "" 0 "" {TEXT -1 27 "With Constant Coefficients." }}{PARA 
19 "" 0 "" {TEXT -1 21 "Math 374    Fall 1999" }}{PARA 19 "" 0 "" 
{TEXT -1 11 "Jay Treiman" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 52 "This is to help illustrate what the solutions are \+
to" }}{PARA 0 "" 0 "" {TEXT -1 57 "linear differential equations with \+
constant coefficients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "
" 0 "" {TEXT 256 46 "It is important that all of your coefficients " }
}{PARA 257 "" 0 "" {TEXT 257 38 "are rational numbers.  You cannot use
 " }}{PARA 258 "" 0 "" {TEXT 258 53 "floating point number coefficient
s in this worksheet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 55 "Restart Maple and load the linalg and DEtools p
ackages." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "restart;\nwith(DEtools)
:\nwith(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 35 "Warning, new definiti
on for adjoint" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition
 for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for \+
trace" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 20 "Distinct real roots." }
}{PARA 0 "" 0 "" {TEXT -1 42 "Here one has a differential equation who
se" }}{PARA 0 "" 0 "" {TEXT -1 42 "charateristic polynomial has distin
ct real" }}{PARA 0 "" 0 "" {TEXT -1 6 "roots." }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 78 "deq1 := diff(y(x),x$3) -2*diff(y(x),x$2) \n     \+
    -5*diff(y(x),x) + 6*y(x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%
deq1G/,*-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"$\"\"\"-F(6$F*-F/6$F-\"\"
#!\"#-F(6$F*F-!\"&F*\"\"'\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 52 "Maple can get you the coefficients and co
nvert them " }}{PARA 0 "" 0 "" {TEXT -1 54 "into the characteristic po
lynomial.  One can then get " }}{PARA 0 "" 0 "" {TEXT -1 43 "the roots
 of the characteristic polynomial." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "coeffs_deq1 := convertAlg(d
eq1,y(x))[1];\npoly_deq1 := sum(coeffs_deq1[i]*x^(i-1),\n             \+
 i=1..vectdim(coeffs_deq1));\nroots_deq1 := [solve(poly_deq1,x)];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,coeffs_deq1G7&\"\"'!\"&!\"#\"\"\"" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*poly_deq1G,*\"\"'\"\"\"%\"xG!\"&*
$)F(\"\"#\"\"\"!\"#*$)F(\"\"$F-F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%+roots_deq1G7%\"\"\"\"\"$!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 54 "There are three distinct roots of this po
lynomial.  We" }}{PARA 0 "" 0 "" {TEXT -1 51 "can now generate the sol
ution and check the answer." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "sol_deq1 := sum(C[i]*exp(roots_deq1
[i]*x),\n               i=1..vectdim(roots_deq1));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%)sol_deq1G,(*&&%\"CG6#\"\"\"F*-%$expG6#%\"xGF*F**&&
F(6#\"\"#F*-F,6#,$F.\"\"$F*F**&&F(6#F6F*-F,6#,$F.!\"#F*F*" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "One can now subst
itute our solution into the differential" }}{PARA 0 "" 0 "" {TEXT -1 
29 "equation to check the answer." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "evalb(eval(subs(y(x)=sol_deq
1,deq1)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "One can also check t
o see if  Maple gets the same solution." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dsolve(deq1,y(x));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&%$_C1G\"\"\"-%$expGF
&F+F+*&%$_C2GF+-F-6#,$F'\"\"$F+F+*&%$_C3GF+-F-6#,$F'!\"#F+F+" }}}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 35 "Real roots of multiplicity above \+
1." }}{PARA 0 "" 0 "" {TEXT -1 51 "Consider the following linear diffe
rential equation" }}{PARA 0 "" 0 "" {TEXT -1 27 "with constant coeffic
ients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "deq2 := diff(y(x)
,x$4) - diff(y(x),x$3)\n          -3*diff(y(x),x$2) + 5*diff(y(x),x)\n
           -2*y(x) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%deq2G/,,
-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"%\"\"\"-F(6$F*-F/6$F-\"\"$!\"\"-F
(6$F*-F/6$F-\"\"#!\"$-F(6$F*F-\"\"&F*!\"#\"\"!" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Now one can find the root
s of the characteristic polynomial" }}{PARA 0 "" 0 "" {TEXT -1 14 "and
 its roots." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 159 "coeffs_deq2 := convertAlg(deq2,y(x))[1];\npoly_deq
2 := sum(coeffs_deq2[i]*x^(i-1),\n              i=1..vectdim(coeffs_de
q2));\nroots_deq2 := [solve(poly_deq2,x)];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%,coeffs_deq2G7'!\"#\"\"&!\"$!\"\"\"\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%*poly_deq2G,,!\"#\"\"\"%\"xG\"\"&*$)F(\"\"#\"
\"\"!\"$*$)F(\"\"$F-!\"\"*$)F(\"\"%F-F'" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%+roots_deq2G7&!\"#\"\"\"F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 50 "One can check the solution Maple give
s against the" }}{PARA 0 "" 0 "" {TEXT -1 25 "roots of this polynomial
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "dsolve(deq2,y(x));\nsol_deq2 := collect(%,exp(x));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,**&%$_C1G\"\"\"-%$expG
F&F+F+*&%$_C2GF+-F-6#,$F'!\"#F+F+*(%$_C3GF+F,\"\"\"F'F+F+*(%$_C4GF+F,F
6)F'\"\"#F6F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)sol_deq2G/-%\"yG6#
%\"xG,&*&,(%$_C1G\"\"\"*&%$_C3GF.F)F.F.*&%$_C4GF.)F)\"\"#\"\"\"F.F.-%$
expGF(F.F.*&%$_C2GF.-F76#,$F)!\"#F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 46 "Note that there are three solutions i
nvolving " }{XPPEDIT 18 0 "exp(x);" "6#-%$expG6#%\"xG" }{TEXT -1 23 ".
  These take the form " }}{PARA 0 "" 0 "" {TEXT -1 25 "that is give in
 the text." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "
" {TEXT -1 13 "Complex roots" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 49 "Now consider the following differential equatio
n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 81 "deq3 := diff(y(x),x$3) + 4*diff(y(x),x$2)\n          \+
+4*diff(y(x),x) + 3*y(x) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%de
q3G/,*-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"$\"\"\"-F(6$F*-F/6$F-\"\"#
\"\"%-F(6$F*F-F8F*F1\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 53 "Now check the roots of the characteristic polynom
ial." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 159 "coeffs_deq3 := convertAlg(deq3,y(x))[1];\npoly_deq3 \+
:= sum(coeffs_deq3[i]*x^(i-1),\n              i=1..vectdim(coeffs_deq3
));\nroots_deq3 := [solve(poly_deq3,x)];" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%,coeffs_deq3G7&\"\"$\"\"%F'\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%*poly_deq3G,*\"\"$\"\"\"%\"xG\"\"%*$)F(\"\"#\"\"\"F)*
$)F(F&F-F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+roots_deq3G7%!\"$,&#!
\"\"\"\"#\"\"\"*&%\"IGF+-%%sqrtG6#\"\"$\"\"\"#F+F*,&F(F+F,F(" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Two of th
e roots are complex numbers.  This gives" }}{PARA 0 "" 0 "" {TEXT -1 
32 "us two trigonometric functions, " }{XPPEDIT 18 0 "sin;" "6#%$sinG
" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos;" "6#%$cosG" }{TEXT -1 7 ". \+
 Here" }}{PARA 0 "" 0 "" {TEXT -1 27 "is the solution from Maple." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 "sol_deq3 := dsolve(deq3,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%)sol_deq3G/-%\"yG6#%\"xG,(*&%$_C1G\"\"\"-%$expG6#,$F)!\"$F-F-*(%$_C2
GF--F/6#,$F)#!\"\"\"\"#F--%$cosG6#,$*&-%%sqrtG6#\"\"$\"\"\"F)F-#F-F:F-
F-*(%$_C3GF-F5FD-%$sinGF=F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 33 "Again one gets the expected form." }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 22 "Multiple complex roots" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Consider what h
appens when there are complex" }}{PARA 0 "" 0 "" {TEXT -1 55 "roots wi
th multiplicity greater than 1.  Here is such a" }}{PARA 0 "" 0 "" 
{TEXT -1 22 "differential equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "deq4 := diff(y(x),x$4) + 2
*diff(y(x),x$3)\n      + 9*diff(y(x),x$2) + 8*diff(y(x),x)\n         +
 16*y(x) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%deq4G/,,-%%diffG6$
-%\"yG6#%\"xG-%\"$G6$F-\"\"%\"\"\"-F(6$F*-F/6$F-\"\"$\"\"#-F(6$F*-F/6$
F-F8\"\"*-F(6$F*F-\"\")F*\"#;\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 23 "Now we check the roots." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "coeffs_d
eq4 := convertAlg(deq4,y(x))[1];\npoly_deq4 := sum(coeffs_deq4[i]*x^(i
-1),\n              i=1..vectdim(coeffs_deq4));\nfactor(poly_deq4);\nr
oots_deq4 := [solve(poly_deq4,x)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%,coeffs_deq4G7'\"#;\"\")\"\"*\"\"#\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%*poly_deq4G,,\"#;\"\"\"%\"xG\"\")*$)F(\"\"#\"\"\"\"\"
**$)F(\"\"$F-F,*$)F(\"\"%F-F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),(
*$)%\"xG\"\"#\"\"\"\"\"\"F(F+\"\"%F+F)F*" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%+roots_deq4G7&,&#!\"\"\"\"#\"\"\"*&%\"IGF*-%%sqrtG6#\"#:\"\"\"
#F*F),&F'F*F+F'F&F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 53 "Since the roots come in pairs, the solutions involve " 
}}{PARA 0 "" 0 "" {TEXT -1 15 "polynomials in " }{XPPEDIT 18 0 "x;" "6
#%\"xG" }{TEXT -1 34 " times tigonometric functions and " }}{PARA 0 "
" 0 "" {TEXT -1 47 "exponentials.  Here is the solution from Maple." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sol_deq4 := dsolve(deq4,y(x)
);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)sol_deq4G/-%\"yG6#%\"xG,**(%$
_C1G\"\"\"-%$expG6#,$F)#!\"\"\"\"#F--%$cosG6#,$*&-%%sqrtG6#\"#:\"\"\"F
)F-#F-F4F-F-*(%$_C2GF-F.F>-%$sinGF7F-F-**%$_C3GF-F.F>FBF>F)F>F-**%$_C4
GF-F.F>F5F>F)F>F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "Exercises" }}{PARA 0 "" 0 "" 
{TEXT -1 2 "1." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Check that the s
olutions obtained above from Maple " }}{PARA 0 "" 0 "" {TEXT -1 57 "ar
e actually the solutions of the differential equations." }}{PARA 0 "" 
0 "" {TEXT -1 51 "(You need to check that they are solutions and that
" }}{PARA 0 "" 0 "" {TEXT -1 44 "there are the appropriate number of l
inearly" }}{PARA 0 "" 0 "" {TEXT -1 23 "independent functions.)" }}}
{PARA 0 "" 0 "" {TEXT -1 2 "2." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "
Solve the differential equation" }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "diff(y(x),`$`(x,4))-2*diff(y(x),`$`(x,3))-7*diff(y(x),`
$`(x,2))+20*diff(y(x),x)-12*y(x) = 0;" "6#/,,-%%diffG6$-%\"yG6#%\"xG-%
\"$G6$F+\"\"%\"\"\"*&\"\"#F0-F&6$-F)6#F+-F-6$F+\"\"$F0!\"\"*&\"\"(F0-F
&6$-F)6#F+-F-6$F+\"\"#F0F:*&\"#?F0-F&6$-F)6#F+F+F0F0*&\"#7F0-F)6#F+F0F
:\"\"!" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "by finding the r
oots of the characteristic " }}{PARA 0 "" 0 "" {TEXT -1 31 "polynomial
 and by using dsolve." }}}{PARA 0 "" 0 "" {TEXT -1 2 "3." }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 48 "Try to solve the following differential e
quation" }}{PARA 0 "" 0 "" {TEXT -1 51 "as it is written and by conver
ting the coefficients" }}{PARA 0 "" 0 "" {TEXT -1 42 "to fractions.  U
se Maple's dsolve command." }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "diff(y(t),`$`(t,2))+.23*diff(y(t),t) = 32;" "6#/,&-%%di
ffG6$-%\"yG6#%\"tG-%\"$G6$F+\"\"#\"\"\"*&$\"#B!\"#F0-F&6$-F)6#F+F+F0F0
\"#K" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 28 "Explain what Maple is doing." }}}}}{MARK "2 0" 16 }
{VIEWOPTS 1 1 0 1 1 1803 }