for the following PDE separate the variables using $\phi(x,y) = X(x)Y(y)$ and find the ODEs satisfied by $X $ and $Y$
PDE: $$ \dfrac{\partial ^2 \phi}{\partial x^2} = \dfrac{\partial^2 \phi}{\partial y^2} + \dfrac{\partial \phi}{\partial y} + \phi $$
Using the substitution I get $$X''Y = Y''X + XY' + XY \implies \dfrac{X''}{X} = \dfrac{Y''}{Y} + \dfrac{Y'}{Y} + 1$$ but wwhere am I to go now?
would it safe to assume that $$\dfrac{X''}{X} = K = \dfrac{Y''}{Y} + \dfrac{Y'}{Y} + 1$$ where $K$ is a constant? so we have the following, $$X'' = KX$$ and $$Y''+Y' = K_1Y$$? with $K_1 = K - 1$