Q:

general solution for non-homogeneous linear PDEs

suppose we have had the general solution of the homogeneous part, and we have a particular solution for the equation. Will the particular solution plus the general solution for homogeneous part be the general solution for the whole equation? I know we have this kind of theorem in ODE, will it hold for PDE?

A:

Yes. This in true in general for any kind of linear equation. If $T$ is a linear operator, the solution of the linear equation
$$
Tx=f
$$
is
$$
x=x_h+x_p
$$
where $x_h$ is the general solution of $Tx=0$ and $x_p$ a particular solution of $Tx=f$. Proof: let $y$ be any solution. Since $T$ is linear, $y-x_p$ is a solution of $Tx=0$.

