For the inhomogeneous steady-state PDE with linear differential operator $D$, $Du(x,y) = Q(x,y)$ with inhomogeneous Dirichlet boundary conditions $u|_{\partial\Omega} = \alpha$, to find the solution, we split up $u(x,y)$ into its particular and complementary parts: $$\tag{1} u(x,y) = u_0(x,y) + v(x,y)$$
where $v(x,y)$ is the particular solution to the PDE subjected to homogeneous boundary conditions $v|_{\partial\Omega} = 0$, and $u_0(x,y)$ is the complementary solution subjected to inhomogeneous boundary conditions $u_0|_{\partial\Omega} = \alpha$.
Is this correct? I know it works this way for ODEs, but wanted to verify if this was still the case for PDEs before proceeding.