I think I am stuck in solving this problem. It involves a wave equation in a circular membrane, so polar coordinates must be used:
$u_{tt}=c²(u_{rr}+{1\over r}u_r+{1\over r²}u_{\theta\theta})$, r<1, t>0
$u(1,\theta,t)=0$
$u(r,\theta,0)=u_0(r,\theta)$
$u_t(r,\theta,0)=v_0(r,\theta)$
where $u_{tt}={\delta²u(r,\theta,t)\over\delta t²}$, being r the radius and $\theta$ the angle. The only extra assumption I can make is that $\Theta(\theta)=\Theta(\theta+2\pi)$.
By separation of variables, I get $u_t(r,\theta,t)=R(r)\Theta(\theta)T(t)$. The only extra assumption I can make is that $\Theta(\theta)=\Theta(\theta+2\pi)$. By substituting in the original PDE:
${T''\over c²T}={R''\over R}+{R'\over rR}+{\Theta''\over r²\Theta}$
And now I can separate:
${\Theta''\over \Theta}=r²{T''\over c²T}-r²{R''\over R}-r{R'\over R}=-\lambda$
$r²{R''\over R}+r{R'\over R}=r²{T''\over c²T}+\lambda=-\mu$
${T''\over c²T}=-{(\mu+\lambda) \over r²}$
So it's time to solve for the angle eigenvalue problem. I get these solutions, which I assume correct as I got them in previous exercises:
${\lambda_0=1; \Theta_0=1}$
${\lambda_n=n²; \Theta_n=A\cos(n\theta)+B\sin(n\theta})$
So now, it's time to solve for the radius. I can't find why, according to my class notes, I must solve this equation:
${r²R''+rR'+(\mu r²-\lambda)}=0$
I don't know how to get to that equation from the previous steps. This may be really simple but, after some time looking into it, I feel like my head is totally biased and I won't find the mistake...
Thank you very much for reading.