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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 341 "This week:  the Laplace T
ransform.  I decided to put off phase plane analysis [5.7] until after
 Laplace transforms because they make it simpler.  Anyhow, you already
 must be somewhat curious how the laplace option in dsolve makes it wo
rk so much better when trig functions are involved (and other times, t
oo), and what the disadvantages are." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 21 "This week's homework:" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "7.1  Read it I guess." 
}}{PARA 0 "" 0 "" {TEXT -1 19 "7.2  p. 363  12, 23" }}{PARA 0 "" 0 "" 
{TEXT -1 23 "7.3  p. 369  12, 29, 37" }}{PARA 0 "" 0 "" {TEXT -1 19 "7
.4  p. 379  11, 24" }}{PARA 0 "" 0 "" {TEXT -1 26 "7.5  p. 387  7, 15,
 25, 37" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
154 "We will do all of the theory of this stuff today and see how it i
s done in Maple the rest of the week, so just relax and try to get the
 big picture today." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with
(inttrans);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "laplace(1,t,
s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "laplace(diff(y(t),t)
,t,s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "laplace(diff(y(t)
,t,t),t,s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f:=convert(p
iecewise(),Heaviside);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "l
aplace(f,t,s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "int(exp(2
*t)*exp(-s*t),t=0..3)+int(exp(-s*t),t=3..infinity);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 70 "restart:with(inttrans): f:=Heaviside(t-3)
+exp(2*t)*(1-Heaviside(t-3));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "laplace(f,t,s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "pl
ot(piecewise(t<1,1,1<t and t<3, t-1, t^2-4), t=0..4);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 9 "7.3 #12  " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "laplace(sin(3*t)*cos(3*t),t,s);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 7 "7.4 #11" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
52 "invlaplace((s^2-26*s-47)/((s-1)*(s+2)*(s+5)), s, t);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 6 "7.5 #1" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "alias(Y(s)=laplace(y(t),t,s));" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 50 "de:=diff(y(t),t,t)-2*diff(y(t),t)+5*y(t)=sin(2
*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "laplace(de,t,s);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "subs(\{y(0)=2,D(y)(0)=4\},
\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(\",Y(s));" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "invlaplace(\",s,t);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "dsolve(\{de,y(0)=2,D(y)(0)=4
\},y(t),laplace);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "subs(
\",de);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "7.5 #13" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 58 "de:=diff(y(t),t,t)-diff(y(t),t)-2*y(t)=-8*cos(t)
-2*sin(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "init:=y(Pi/2)
=1, D(y)(Pi/2)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "dsolve
(\{de,init\},y(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simp
lify(\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "dsolve(\{de,in
it\},y(t),laplace);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with
(plots);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 187 "pointplot(\{[0
,1],[1,-1],[3,0],[4,-3]\},axes=BOXED);\npointplot(\{seq([n,sin(n/10)],
n=0..30)\});\npoints:= \{ seq([T^2,T],T=0..40) \}:\npointplot(points);
\npointplot(points,coords=polar,color=red);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}}{MARK "33 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }