Using the Wronskian and Indefinite Integrals, I can write the solution to the general one dimensional second order linear non-homogeneous differential equation $$ y''+p(x)y'+q(x)y=g(x) \\ y^*(x)= y_2(x)\int \dfrac{y_1(x)g(x)}{W(x)} dx \\ y_1(x)\int \dfrac{y_2(x)g(x)}{W(x)} dx \\ W(x) = e^{-\int p(x) dx}= y_2'(x)y_1(x)-y_1'(x)y_2(x) \\ y_1''+p(x)y_1'+q(x)y_1=y_2''+p(x)y_2'+q(x)y_2=0 $$
however I am puzzled on how to generalize this prescribed solution to its three dimensional equivalent. I assume the general 3D equivalent is \begin{align} \nabla^2 y+\boldsymbol{p}\cdot \boldsymbol{\nabla}y+qy=g \end{align} where $\boldsymbol{p}$ may have to be curl free.
Does there exist a known solution to the general solution to the three dimensional linear second order partial differential equation using the Wronskian and Indefinite Integrals?