Solved Convert Van Der Pol Oscillator into system of first order ODEs

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Introduction : Van Der Pol Oscillator is a second order ODE. This equation is converted into a system of first order ODEs.

Question : Convert Van Der Pol Oscillator

\[\frac{d^2x}{dt^2}-3\left(1-x^2\right)\frac{dx}{dt}+x=0 , x(0) =1, x'(0) =0\]

into a system of first order ODEs.

Solution :

Van Der Pol Oscillator equation

\[\frac{d^2x}{dt^2}-3\left(1-x^2\right)\frac{dx}{dt}+x=0\]

with initial conditions

\[x\left(0\right)=1, x^\prime\left(0\right)=0\]

can be converted into a system of first order ODEs if we put

Let \[y_1=x ,y_2=\frac{dx}{dt}\]

then we have

\[\frac{dy_1}{dt}=\frac{dx}{dt}=y_2\]

or

\[\frac{dy_1}{dt}=y_2 \] (1)

on the other hand

\[\frac{dy_2}{dt}=\frac{d^2x}{dt^2}=3\left(1-x^2\right)\frac{dx}{dt}-x\]

or

\[\frac{dy_2}{dt}=3\left(1-y_1^2\right)y_2-y_1\] \[\frac{dy_2}{dt}\]

\[\frac{dy_2}{dt}=3\left(1-y_1^2\right)y_2-y_1 \] (2)

subject to Initial Conditions

\[ y_1\left(0\right)=2,\ y_2\left(0\right)=0\] (3)

equations (1) and (2) form a system of first order ODEs with initial conditions given by equation (3) .


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