Consider the following partial differential equation: $$ \left(\frac{\partial~f(x,y)}{\partial x}\right)^2-f(x,y)=0~. $$
I have the following doubt, given that only one partial derivative is present can this equation be, somehow, turned into an ODE? My train of thought was: for a specific value o f $y$, the above equation would be for all means and purposes an ode and my guess is that the dependence in $y$ would be given by the initial condition $f(0,y)=g(y)$ for some function $g(y)$. Does this make sense? If so, would this be the case for every PDE with partial derivatives with respect to only one variable?