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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 75 "Example 2, Chapter 6.4:  Discontinuous Forcing functions (Probl
em 9, p 322)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "g:=(t/2)*(H
eaviside(t)-Heaviside(t-6))+3*Heaviside(t-6);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 26 "de:=diff(y(t),t$2)+y(t)=g;" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 232 "We could solve th
is directly using dsolve, but it may be nice to be able to check the s
teps we take when we're manually solving this differential equation.  \+
Below, we take the Laplace transform and solve for Y(s) as we did in c
lass." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Y:=laplace(de,t,s);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Y1:=subs(\{y(0)=0,D(y)(0)=1
\},Y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Y2:=solve(Y1,lapl
ace(y(t),t,s));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "H:=1/(s^
2*(s^2+1)); #This is the H(s) as in class" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 21 "convert(H,parfrac,s);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "invlaplace(H,s,t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 123 "Now that we have double checked our work, we'll solve it direc
tly.  Note the use of \"method=laplace\" in the dsolve command." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Ys:=dsolve(\{de,y(0)=0,D(y)(
0)=1\},y(t),method=laplace);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 57 "Now we'll compare the forcing function \+
g to the solution:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "plot(
[g,rhs(Ys)],t=0..15,color=black,linestyle=[1,2],title=\"Forcing versus
 Response\");" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 106 "We'll take a look at the derivatives of the solution-
 Which derivatives are continuous, and which are not?" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "yy:=rhs(Ys):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 39 "plot(diff(yy,t$3),t=0..40,color=black);" }}}}
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