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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Week 10-11 exercises." }}
{PARA 0 "" 0 "" {TEXT -1 18 "7.4  5, 10, 20, 31" }}{PARA 0 "" 0 "" 
{TEXT -1 21 "7.5  11, 12, 20, 21  " }}{PARA 0 "" 0 "" {TEXT -1 21 "7.6
  7, 9, 25, 26, 28" }}{PARA 0 "" 0 "" {TEXT -1 18 "7.7  7, 10, 15, 29
" }}{PARA 0 "" 0 "" {TEXT -1 25 "7.8  3, 7, 13, 21, 25, 35" }}{PARA 0 
"" 0 "" {TEXT -1 22 "7.9  1, 11, 12, 21, 22" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 34 "f:=t->piecewise(t<0,0,t<2*Pi,t,0);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(f(t),t=0..8);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "convert(f(t),Heaviside);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "g:=t->sum(f(t-2*Pi*n),n=0..i
nfinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "test:=convert(
g(t),Heaviside);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "with(in
ttrans);laplace(test,t,s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
18 "invlaplace(\",s,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "
h:=t->sum(f(t-n*2*Pi),n=0..8);\nplot(h(t),t=0..25,discont=true,color=p
urple, scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 16 "discont(h(t),t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "This i
s really wierd:  Maple knows the discont. of h but plot won't use them
.  If the discont. are at the integers then it does." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 29 "de:=diff(y(t),t,t)+y(t)=h(t);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "sol:=dsolve(\{de,y(0)=0,D(y)(0)=0\}
,y(t),laplace);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot(rhs
(sol),t=0..2*8*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "deg:
=\{diff(y(t),t,t)+y(t)=test,y(0)=0,D(y)(0)=0\};" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 25 "dsolve(deg,g(t),laplace);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 27 "pdesolve(deg,g(t),laplace);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 13 " Convolution." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 24 "restart; with(inttrans);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 37 "laplace(int(f(t-u)*g(u),u=0..t),t,s);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 19 "This is called the " }{TEXT 256 19 "convo
lution theorem" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 18 "invlaplace(\",s,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
52 "de:=a*diff(y(t),t,t)+b*diff(y(t),t)+c*y(t)=Dirac(t);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "laplace(de,t,s);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 27 "subs(\{y(0)=0,D(y)(0)=0\},\");" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solve(\",laplace(y(t),t,s));
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "This is the " }{TEXT 257 17 "
transfer function" }{TEXT -1 40 " of L[y].  The inverst transform is t
he " }{TEXT 258 15 "impulse respons" }{TEXT -1 4 "e.  " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "assume(v>0);int(Dirac(v-u)*f(u), u=
0..v);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "Now combine this with \+
the convolution theorem:  If  p(t)  is the impulse response, then the \+
solution of  L[y]=g(t)  is  p*g(t)." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "a:=1;b:=2;c:=3;de;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "dsolve(\{de,y(0)=0,D(y)(0)=0\},y(t),laplace);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "dsolve(\{a*diff(y(t),t,t)+b*
diff(y(t),t)+c*y(t)=g(t),y(0)=0,D(y)(0)=0\}, y(t), laplace);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "See?  Maple knows this too." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "dsolve(\{a*diff(y(t),t,t)+b*
diff(y(t),t)+c*y(t)=g(t),y(0)=0,D(y)(0)=0\}, y(t));" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 120 "Leave out the laplace and you get the variatio
n of parameters solution.  For higher order equations this is a real m
ess." }}}}{MARK "34 0 0" 120 }{VIEWOPTS 1 1 0 3 2 1804 }