{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 }{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 }}
{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 15 "MATH 308 Honors" }}{PARA 0 "" 0 "" {TEXT 
-1 11 "Section 200" }}{PARA 0 "" 0 "" {TEXT -1 11 "Spring 2000" }}
{PARA 0 "" 0 "" {TEXT -1 10 "P. Yasskin" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 22 "Exam 2 Maple Solutions" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with
(plots):with(DEtools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "#2" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "deq:=diff(y(t),t,t)+b*diff(y
(t),t)+5*y(t)=cos(w*t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "#2a" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "critically damped when " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "b^2-20=0;" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 2 "or" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "b
=sqrt(20);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "overdamped when the
re are two real roots." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "b
>sqrt(20);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "underdamped when th
ere are two complex roots" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
11 "b<sqrt(20);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "assume(b
<sqrt(20),b>0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "#2b" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "sol:=dsolve(deq,y(t));" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 3 "#2c" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 24 "deq2:=subs(b=2,w=1,deq);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "inits:=y(0)=3/5,D(y)(0)=1/10;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 32 "sol2:=dsolve(\{deq2,inits\},y(t));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "combine(sol2);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 24 "plot(rhs(sol2),t=0..12);" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 3 "#2d" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 133 "For large t, the exponentials drop out.  This can b
e accomplished by setting _C1=0 and _C2=0 in sol.  So the steady state
 solution is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Ys:=subs(_C
1=0,_C2=0,rhs(sol));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "We want t
o write this as:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Ys2:=ex
pand(A*cos(w*t-p));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "So we equa
te and solve" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "solAp:=solv
e(identity(Ys2=Ys,t),\{A,p\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 21 "As:=subs(solAp[1],A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 21 "ps:=subs(solAp[1],p);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "#2e
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Ass:=seq(subs(b=k,As),k
=1..4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "plot([Ass],w=0..
10, color=[red, blue, green, magenta]);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 3 "#2f" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "dA:=diff
(As,w);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "wsol:=solve(dA=0
,w);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "wR:=wsol[2];" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "#2g" }}{PARA 0 "" 0 "" {TEXT -1 25 
"There is a resonance when" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "-2*b^2+20>0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "or " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "b<sqrt(10); evalf(%);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "This agrees with the plot. There i
s a max for b=1,2,3 but not for b=4." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 3 "#2h" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(w=wR,As);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "AsR:=simplify(%);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "#2i" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "plot(AsR,b=0..3.5);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }