Find a formal solution $u(x; y)$ by using Fourier series.
(Hint: In two dimensions the basis functions have one of the forms $\sin(ax) \sin(by)$, $\sin(ax) \cos(by)$ and $\cos(ax) \cos(by)$, with appropriate values for $a$ and $b$).
I am not able to use this and get the solution: Can someone please help.
try the solution of the form $u = \sin mx \sin ny $. putting $u$ in the equation gives $Lu = u -3u_{xx} - u_{yy} = (1 + 3m^2 + n^2)u$ so that $$L^{-1}\left( \sin mx \sin ny \right) = \dfrac{1}{1 + 3m^2 + n^2} \sin mx \sin ny$$
for your problem you need to compute $$L^{-1}\left(\sin 2x \sin 3y - \sin 5x \sin y \right).$$