I am trying to solve the van der pol oscillator, analytically using multiple time scaling and perturbation method.
While doing so, I came across the following pde which looks quite trivial but I can't apply the analytical methods that I was taught in pde module.
$ \frac{\partial^2 x}{\partial^2 t}-\omega_0^2 x=0$
x is a function of t and $\tau$, so x=x(t,$\tau$) and $\omega_0$ is a constant.
A=1, B=C=D=E=G=0, F=$\omega_0^2$
so $\frac{dy}{dx}=\frac{B}{A} = 0$
$\implies$y(t)=constant=c=$\xi(t)$ but this choice of $\xi$ gives the jacobian matrix 0 for any $\eta(t)$. But we want the jacobian matrix to be non zero. So how do I choose the function for $\xi(t)$ and $\eta(t)$?
And what is a particular solution to the following pde?
$ \frac{\partial^2 x}{\partial^2 t}-\omega_0^2 x=2 A_0^3\omega_0 sin(3 \omega_0 t)+cos(\omega t)$
where $\omega_0$ and $\omega$ are not equal and $A_0$ =$\sqrt{\frac{1}{1+a_0 \exp(-\tau)}}$ exp(-i$\alpha_0$).
$\alpha_0$=$arcsin$($\frac{-B}{\sqrt{A^2\omega_0^2+B^2}})$ and $a_0$=$\frac{4\omega_0^2}{A^2\omega_0^2+B^2}$-1
where A and B are constants of Boundary condition.