Find the general solution of homogenous PDE

Find the solution of the equation $$\partial^2{z/\partial{x^2}} + \partial^2{z/\partial{y^2}} = e^{-x}cosy$$.

I am able to find the Complementary Function as $$z_c = φ_1(y + ix) + φ_2(y - ix)$$. Please explain how to find the Particular Integral (P.I.).

For non-homogenous PDE, we can find P.I. for $$f(D,D')z = e^{ax+by}.V$$ as: $$z_p = \frac{1}{f(D,D')}e^{ax+by}.V = e^{ax+by}.\frac{1}{f(D + a,D'+b)}.V$$

Can we use the same method for homogenous equations also ?

Let's substitute the ansatz $$z_p=f(x)\,e^{-x}\cos y$$ in the PDE. Then, a straightforward calculation yields $$[f''(x)-2f'(x)]\,e^{-x}\cos y = e^{-x}\cos y \implies f''(x)-2f'(x)=1. \tag{1}$$ The general solution to $$(1)$$ is $$f(x)=c_1+c_2e^{2x}-\frac{x}{2}, \tag{2}$$ where $$c_1$$ and $$c_2$$ are arbitrary constants. Since we want a particular solution to the PDE, not the most general one, we may choose $$c_1=c_2=0$$, finally obtaining $$z_p=-\frac{1}{2}xe^{-x}\cos y. \tag{3}$$