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 \(u(2,\theta)=-8\cos(3\theta)+64\cos(5\theta)\)

Answer 1

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

Answer 2

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

Answer 3

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...