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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 93 "This is an examination of the differential equation, y''+6y=Dir
ac(t), with y(0)=0 and y'(0)=0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 46 "Before we begin, let's get a general solu
tion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "DE:=diff(y(t),t$2)+6*y(t)=Dirac(t);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%#DEG/,&-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F-\"\"#\"\"
\"*&\"\"'F2F*F2F2-%&DiracGF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "dsolve(DE,y(t),method=laplace);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/-%\"yG6#%\"tG,&*&-F%6#\"\"!\"\"\"-%$cosG6#*&\"\"'#F-\"\"#F'F-F-F-*
&#F-F2F-*(F2F3--%\"DG6#F%F+F--%$sinGF0F-F-F-" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "This form of the solut
ion would say that, with our initial conditions, the solution is y(t)=
0.  But is that correct?" }}{PARA 0 "" 0 "" {TEXT -1 22 "Let's try it \+
manually:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 15 "with(inttrans):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "readlib(laplace):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "Lde:=laplace(DE,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$LdeG/,(*&%\"sG\"\"\",&*&F(F)-%(laplaceG6%-%\"yG6#%\"tGF2F(F)F
)-F06#\"\"!!\"\"F)F)--%\"DG6#F0F4F6*&\"\"'F)F,F)F)F)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 35 "Lsol:=solve(Lde,laplace(y(t),t,s));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%LsolG*&,(*&%\"sG\"\"\"-%\"yG6#\"\"!
F)F)--%\"DG6#F+F,F)F)F)F),&*$)F(\"\"#F)F)\"\"'F)!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Sol:=invlaplace(Lsol,s,t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SolG,&*&-%\"yG6#\"\"!\"\"\"-%$cosG6
#*&\"\"'#F+\"\"#%\"tGF+F+F+*&#F+F0F+*(F0F1,&--%\"DG6#F(F)F+F+F+F+-%$si
nGF.F+F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 95 "This time with our initial conditions, Maple says the sol
ution is y(t)=(1/sqrt(6))sin(sqrt(6)t)" }}{PARA 0 "" 0 "" {TEXT -1 82 
"  But is that correct?  This doesn't satisfy the initial conditions, \+
since y'(0)=1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 96 "For our next trial, let's take Dirac(t-w), where w>0, and
 then take the limit as w goes to zero." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assume(w>0);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "DE2:=diff(y(t),t$2)+6*y(t)=D
irac(t-w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DE2G/,&-%%diffG6$-%\"
yG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&\"\"'F2F*F2F2-%&DiracG6#,&F-F2%#w|irG!
\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Sol2:=dsolve(DE2,y(
t),method=laplace);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Sol2G/-%\"yG
6#%\"tG,(*&-F'6#\"\"!\"\"\"-%$cosG6#*&\"\"'#F/\"\"#F)F/F/F/*&#F/F4F/*(
F4F5--%\"DG6#F'F-F/-%$sinGF2F/F/F/*&F8F/*(-%*HeavisideG6#,&F)F/%#w|irG
!\"\"F/F4F5-F?6#*&F4F5FEF/F/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "limit(rhs(Sol2),w=0);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,(*&-%\"yG6#\"\"!\"\"\"-%$cosG6#*&\"\"'#F)\"\"#%\"tGF)F)F)*&#F)F
.F)*(F.F/--%\"DG6#F&F'F)-%$sinGF,F)F)F)*&F3F)*(-%*HeavisideG6#F1F)F.F/
F9F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 109 "Now the solution seems to make more sense:  The solution
, with our initial conditions, is the same as before," }}{PARA 0 "" 0 
"" {TEXT -1 91 "but the Heaviside function is included.  Does this sol
ution satisfy the initial conditions?" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "y2:=(1/sqrt(6))*Heavisi
de(t)*sin(sqrt(6)*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2G,$*&#\"
\"\"\"\"'F(*(-%*HeavisideG6#%\"tGF(F)#F(\"\"#-%$sinG6#*&F)F/F.F(F(F(F(
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "diff(y2,t);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"'F&*(-%&DiracG6#%\"tGF&F'#F&
\"\"#-%$sinG6#*&F'F-F,F&F&F&F&*&-%*HeavisideGF+F&-%$cosGF1F&F&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(t=0,y2);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"'F&*(-%*HeavisideG6#\"\"!F&F'#F&
\"\"#-%$sinGF+F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 27 "And how did Maple do that??" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "So, what is the correct \+
answer??   One way to interpret the second answer (the sine function w
ithout the Heaviside) is" }}{PARA 0 "" 0 "" {TEXT -1 118 "that solving
 our differential equation, with zero initial conditions, is the same \+
as solving y''+6y=0, y(0)=0, y'(0)=1" }}{PARA 0 "" 0 "" {TEXT -1 48 "s
ince the Dirac function has \"unit impulse\"...  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "18 6 0" 0 }{VIEWOPTS 1 
1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }