particular integral in partial differential equation Solve the partial differential equation
$$\left[D^2+{D^\prime}^2\right]z=\cos mx\cdot\cos ny.$$
I have a problem in finding particular integral where $\displaystyle\frac\partial{\partial x}=D$ and $\displaystyle\frac\partial{\partial y}=D^\prime$.

$$\frac{\partial^2z}{\partial x^2}+\frac{\partial^2z}{\partial y^2}=\cos (mx)\,\cos (ny)\qquad (1)$$
The particular solution of (1) is sought in the form
$$z=A\,\cos (mx)\,\cos (ny)$$
Substitution this in (1) yields
$$-A\, {{n}^{2}} \cos{\left( m x\right) } \cos{\left( n y\right) }-A\, {{m}^{2}} \cos{\left( m x\right) } \cos{\left( n y\right) }=\cos{\left( m x\right) } \cos{\left( n y\right) }$$
Then $$A=-\frac{1}{n^2+m^2},$$
$$z_p=-\frac{\cos (mx)\cdot\cos (ny)}{n^2+m^2}$$
