Consider this problem
$$ \begin{cases} -\Delta u=10 \hspace{6mm} \mbox{in} \hspace{6mm} \Omega \\ u=0 \hspace{6mm}\mbox{on}\hspace{6mm}\Gamma_d \\ \frac{\partial u}{\partial n}=-\sqrt{4x^2+64y^2} \hspace{3mm} \mbox{on} \hspace{3mm} \Gamma_N \end{cases} $$ $$ \Omega=\{(x,y)\mid x^2+4y^2<16\}\\ \Gamma_d=\{(x,y)\in \partial \Omega\mid y\le 0\} \\ \Gamma_N=\{(x,y)\in \partial \Omega\mid y> 0\} $$
I want to convert above poisson equation to laplace equation. I know that the exact solution of the above equation is $$ u_e(x,y)=16-x^2-4y^2 $$ My idea for converting it would be to consider $u=u_0 + \widetilde{u}$ and $u_0=\frac{-5}{2}x^2-\frac{5}{2}y^2$. But I am not exactly sure of side effects of substituting this into the above equation. Will it change the boundary conditions? How so?