I have the following PDE: $$C \frac{\partial^2 u(x, t)}{\partial x^2} + f(x,t) = \rho\frac{\partial^2 u(x,t)}{\partial t^2}$$ Where $\left. f(x,t)\right|_{x>0} = 0$. The domain is $(x, t) \in [0, \infty) \times [0, \infty)$.
How do I approach this? Can this be split into two PDEs, one at $x=0$ and the other at $x>0$, and then equate the boundary conditions at $x=0$? But if you split it into two, would the spatial derivative exist around $x=0$?