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) .