All Questions

Consider $u = u(\phi,\psi)$ where $\phi = \phi(x)$ and $\psi =\psi(x)$ are both analytic function. The following equation $$\partial_x u - u\partial_x (\phi-\psi)=0$$ has a trivial solution $u(\phi,\...

I would like to know if we can find a real function $v(x)$ and a complex function $f(x)$, such that they solve the following differential equation (with $\alpha$ a complex, $0<Re(\alpha)<1$): $$...