Solve inhomogeneous Laplace Equation (Poisson's Equation)

Question

First of all, it's not a homework or exams. I want to learn more about this PDE. So, if you don't mind please help me to solve this problem and i'll appreciate it.

Given problem=

$u_{xx}+u_{yy}=x$

$x^2+y^2<3$

$u\left(\sqrt{3},\theta\right)=e^\theta$

Answer 1

Let $v=x^3/6$. Then $$ v_{xx}+v_{yy} = x. $$ This reduces the problem to solving for $w=u-v$ satisfying $$ w_{xx}+w_{yy}=0 \\ w(\sqrt{3},\theta)= e^{\theta}-r^3\cos^2(\theta)/6. $$ And $u=w+v = w+x^3/6$ is your solution.