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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 8 "ODE Tour" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 12 "with(plots):" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Legendre's
 Equation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "ode1 := (1-t^2)
*diff(y(t),t$2)-2*t*diff(y(t),t)+2*y(t)=0;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 18 "dsolve(ode1,y(t));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "dsolve(\{ode1,y(0)=1,D(y)(0)=0\},y(t));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "lprint(%);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 37 "y1 := t -> -1/2*t*ln(-(t+1)/(t-1))+1;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(y1(t),t=-0.9 .. 0.9);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "dsolve(\{ode1,y(0)=0,D(y)
(0)=1\},y(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "y2 := t -
> t;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(\{y1(t),y2(t)
\},t=-0.9 .. 0.9);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Bessel's \+
Equation of order n=1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "n :
= 1:\node2 := diff(y(t),t$2)+diff(y(t),t)/t+(1-n^2/t^2)*y(t)=0;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dsolve(ode2,y(t));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot( \{BesselJ(1,t),BesselY
(1,t)\},t=0.1 .. 5);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "Duffing
's Equation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:\nwit
h(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "ode3 := diff(y
(t),t$2)+y(t)+y(t)^3=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "
dsolve(ode3,y(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "sol:=
 dsolve(\{ode3,y(0)=0,D(y)(0)=1\},y(t),type=numeric,\noutput=listproce
dure);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "y2 := subs(sol,y(
t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "for i from 0 to 300
 do\n  val[i] := y2(i*0.1);\nend do:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "p1 := plot( [[0.1*n,val[n]] $n=0..300],x=0..30):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "p2 := plot( sin(x), x=0..30,
color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display(\{p
1,p2\});" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Pendulum Equation" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:\nwith(plots):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ode4 := diff(y(t),t$2)+y(t)
=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "dsolve(\{ode4,y(0)=0
,D(y)(0)=1\},y(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "ode5
 := diff(y(t),t$2)+sin(y(t))=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 79 "sol := dsolve(\{ode5,y(0)=0,D(y)(0)=1\},y(t),type=numeric,\nou
tput=listprocedure):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "y5 \+
:= subs(sol,y(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "for i
 from 0 to 300 do\n  val[i] := y5(i*0.1);\nend do:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 48 "p5 := plot( [[0.1*n,val[n]] $n=0..300],x=0.
.30):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "p4 := plot( sin(x)
, x=0..30,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "d
isplay(\{p4,p5\});" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "Unforced \+
Mass-Spring System" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restar
t:\nwith(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ode6 :=
 m*diff(y(t),t$2)+b*diff(y(t),t)+k*y(t)=0;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 18 "dsolve(ode6,y(t));" }}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 21 "Case 1: (overdamped) " }{XPPEDIT 18 0 "0 < b^2-4*km;" "6#
2\"\"!,&*$%\"bG\"\"#\"\"\"*&\"\"%F)%#kmGF)!\"\"" }{TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "b :=1; m:=1; k:=0.1; b^2-4*k
*m;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "dsolve(\{ode6,y(0)=0
,D(y)(0)=1\},y(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "lpri
nt(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "y := t -> -1/3*15
^(1/2)*exp(-1/10*(5+15^(1/2))*t)+1/3*15^(1/2)*exp(1/10*(-5+15^(1/2))*t
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(y(t),t=0..100);
" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Case 2: (underdamped) " }
{XPPEDIT 18 0 "b^2-4*km < 0;" "6#2,&*$%\"bG\"\"#\"\"\"*&\"\"%F(%#kmGF(
!\"\"\"\"!" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 
"unassign(y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "b := 0.1; \+
m:=1; k:=1; b^2-4*k*m;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "o
de6 := m*diff(y(t),t$2)+b*diff(y(t),t)+k*y(t)=0;" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 37 "dsolve(\{ode6,y(0)=0,D(y)(0)=1\},y(t));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "lprint(%);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 62 "y := t -> 20/399*399^(1/2)*exp(-1/20*t)*s
in(1/20*399^(1/2)*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "pl
ot(y(t),t=0..100);" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Case 3: (
critically damped) " }{XPPEDIT 18 0 "b^2-4*km = 0;" "6#/,&*$%\"bG\"\"#
\"\"\"*&\"\"%F(%#kmGF(!\"\"\"\"!" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 38 "b := 1; k := 1; m := 0.25; b^2 -4*k*m;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "unassign(y);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ode6 := m*diff(y(t),t$2)+b*diff(y(t
),t)+k*y(t)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "dsolve(\{
ode6,y(0)=0,D(y)(0)=1\},y(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 10 "lprint(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "y := \+
t -> exp(-2*t)*t;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(y
(t),t=0..10);" }}}}}}{MARK "5" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
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