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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 189 "Example from class:  Convert a second order ODE to a system of
 first order.  Plot the results of y' vs. y, y and y' versus t.  We wi
ll also plot the solution overlaying the direction field." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta
rt;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "dU:=diff(u(t),t)=v(t);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "dV:=diff(v(t),t)=-v(t)-1.25*
u(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "DEplot([dU,dV],[u
(t),v(t)],t=-1..6,u=-4..4,v=-4..4,[[u(0)=3,v(0)=1]],stepsize=0.05,line
color=black);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 141 "Now we'll plot u (or y) versus t and v (or y') versus
 t on the same graph.  The position (y) will be blue, and the velocity
 (y') will be red:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 114 "A:=DEplot([dU,dV],[u(t),v(t)],t=0..15,u=-4
..4,v=-4..4,[[u(0)=3,v(0)=1]],stepsize=0.05,scene=[t,u],linecolor=blue
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "B:=DEplot([dU,dV],[u
(t),v(t)],t=0..15,u=-4..4,v=-4..4,[[u(0)=3,v(0)=1]],stepsize=0.05,scen
e=[t,v],linecolor=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "
with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "display(\{A
,B\},title=\"Plot of position and velocity\");" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 8 }{VIEWOPTS 1 1 0 1 1 
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