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**# This is a sample worksheet for your homework assignment. Use commond `?dsolve[piecewise]' to find out how to modify this worksheet for your piecewise defined equation. **

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**with(DEtools):with(plots):**

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**mu:=10:**

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**sys:=D(x)(t)=mu*(y(t)-x(t)^3/3+x(t)),D(y)(t)=-x(t)/mu:# The van der Pol equation as a system of 1st order equations.**

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**solu:= dsolve({sys,x(0)=1,y(0)=.4},{x(t),y(t)},type=numeric):**

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**a:=odeplot(solu, [x(t),y(t)],0..20,numpoints=100):**

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**b:=plot([x, x^3/3-x, x=-3..3],-3..3,-1..1,color=black):**

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**display({a,b});**

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**xp:=odeplot(solu,[t,x(t)],0..50,color=red):yp:=odeplot(solu,[t,y(t)],0..50,color=green):**

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**display({xp,yp});**

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**phaseportrait([D(x)(t)=mu*(y(t)-x(t)^3/3+x(t)),D(y)(t)=-x(t)/mu], \
[x(t),y(t)],t=0..20,[[x(0)=1,y(0)=0]],stepsize=.05, \
scene=[x(t),y(t)],linecolour=sin(t*Pi/2)); # Phaseportrait.**

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**ode:=diff(y(t),t$2)+mu*(y(t)^2-1)*diff(y(t),t)+y(t)=0: # You can solve it as a 2nd order equation as well. **

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**ic:=y(0)=1,D(y)(0)=0:**

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**solu2:=dsolve({ode,ic},y(t),type=numeric):**

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**solu2(2); # Check the solution as a given time t=2.**

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**odeplot(solu2,[t,y(t)],0..50,color=black);**

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