# How to solve $y(x) y'''(x)=f(x)$

I have a PDE of the form $$\partial_t y(x,t)+\partial_x(y(x) y'''(x)-f(x))=0$$, where $$f(x)=\cos(x)$$.

Suppose a stable equilibrium exists, and I want to find the steady-state solution $$y(x) y'''(x)=f(x)$$ and I expect the physical solution to be periodic (as the source term is also periodic). How do I solve this equation (or 3-order nonlinear ODE in general) numerically (or analytically) in Scipy/Matlab/Mathematica?

For the boundary conditions, for the original problem I can just specify $$y(x,0)$$ and evolve the system, and the steady-state solution can be obtained for large $$t$$. But if I want to solve the steady-state ODE directly, do I need to figure out three equations for the boundary conditions for it to be solvable?

• If you assume that your solution is periodic over the space domain, could you write $y(x)$ and $f(x)$ as a Fourier expansion, work out the derivative and multiplication in the LHS and apply the Fourier integral left and right? If you assume $f(x)$ to have a Fourier expansion, that would leave only the Fourier expansion coefficients at the RHS. This transforms your non-linear ode in a large set of equations for the Fourier coefficients for $y(x)$. But as I write this down, I realise that this might not necessarily make things easier.... Jul 2 '19 at 6:27
• Does y depend on t or not? Jul 3 '19 at 8:04
• To solve it numerically, you can use finite different scheme or finite element scheme based on B-Splines (since you need higher continuities). Jul 3 '19 at 13:39