Laplace's equation in polar coordinates

$$\Delta u(r,\theta)=0$$

inside the region$$(0\leq\theta

Question

Laplace's equation in polar coordinates

$$\Delta u(r,\theta)=0$$

inside the region$$(0\leq\theta < \pi/2$$$$0\leq r \leq 2$$where the boundary $\theta=0$ is insulated and $u(r,\pi /2)=0$.

Give the radial solution with $u(2,\theta)=-8\cos(3\theta)+64\cos(5\theta)$

richyw · Answer

I used separation of variables $u(r,\theta)=G(r)\Phi(\theta)$ and then $$\frac{r}{G}\left(\frac{\partial G}{\partial r}+r\frac{\partial^2 G}{\partial r^2}\right)=-\frac{\partial^2\Phi}{\partial \theta^2}=-\lambda$$where $\lambda$ is a constant

richyw · Answer

solving for $\Phi(\theta)$ I get$$\lambda_n=(2n-1)^2$$$$\Phi_n(\theta)=\cos((2n-1)x)$$

richyw · Answer

but when I try to solve for G(r), by assuming $G(r)=r^m$, I end up with$$m^2+(2n-1)^2=0$$which doesn't make sense...