At the end of the problem you will find a complete list of the notations that you will need to enter and how to enter them.

A model for a vibrating drum head with radius c (clamped), initial displacement f, and initial velocity g is given by

We will also require that u be bounded inside the disk $0\leq r\leq c$ and that $u(r,\theta,t) = u(r,\theta+2m\pi)$.

Separation
Suppose that $u(r,\theta,t) = R(r)\Theta(\theta)T(t)$ is separable. Substitute this into the PDE:

Divide both sides of this equation by $a^2R\Theta T$:

We know have two separated equations (your answers should not have fractions with a dependent variable in the denominator)

The equation on the left in R and $\Theta$ is known and Hemholtz's equation, which we will now separate. Using Hemholtz's equation, keep all of the terms with $\Theta$ on the left side, and move the term with $\Theta^{\prime\prime}$ to the right:

Now divide both sides by $-\frac{1}{r^2} {\rm R}\Theta$:

Now we have separated the equations in R, $\Theta$ and T. If you have not already done so, rewrite the equation in R so that there are no fractions and the coefficient on $R^{\prime\prime}$ is positive.
Equation in R: $=0$
Equation in $\Theta$: $=0$
Equation in T: $=0$

Solving the Separated ODEs
Solve for $\Theta$:
From the conditions on the PDE above, we know that $\Theta(\theta) = \Theta\Big(\theta +$ $\Big)$
There is only one set of solutions of the ODE in $\Theta$ that are periodic with period $2\pi$ (use the coefficients Am and Bm, your answer should be in terms of m, Am, and Bm):
$\Theta(\theta) =$
Solve for R:
Therefore $\nu =$ and the ODE in R is now
$=0$
which is Bessel's equation of order m. There are two linearly independent of this equation, and since n is an integer the solutions are $J_m$ and $Y_m$ - Bessel functions of the first and second kind respectively. Recall that we require R to be bounded in the disk $0\leq r\leq c$, therefore we can reject the solution $Y_m$.
Therefore we have one solution R(r)$= J_m\Big($ $\Big)$. From the boundary conditions we must have
R$\Big($$\Big) = J_m\Big($ $\Big) =$
There are infinitely many solutions of $J_m\left(x\right) 0$ for each m, we will call these solutions $\alpha_{m,n}$, so that $\lambda =$
Solve for T:
The ODE in T can now be written in terms of $\alpha_{m,n}$ as
with $\alpha_{m,n} > 0$, so the solution is (using $A_{m,n}$ and $B_{m,n}$ as coefficients)
$T(t) =$

Series

Now that we have the separated equations solved we put them back together as a series
$u(r,\theta,t) = \displaystyle\sum\limits_{m=0}^\infty\sum\limits_{n=1}^\infty T_m\Theta_{m,n}R_{m,n}$
$\hskip 35pt =\displaystyle\sum\limits_{m=0}^\infty\sum\limits_{n=1}^\infty \Big($$\Big)J_m\Big($$\Big)$

Notation

$\displaystyle\lambda = {\rm lambda}$

$\displaystyle\nu = {\rm nu}$

$\displaystyle\frac{d{\rm R}}{d{\rm r}} = {\rm R'}$

$\displaystyle\frac{d{\rm R}}{d{\rm r}} = {\rm R''}$

$\displaystyle\frac{d{\rm T}}{d{\rm t}}={\rm T'}$

$\displaystyle\frac{d{\rm T}}{d{\rm t}}={\rm T''}$

$\displaystyle\theta={\rm theta}$

$\displaystyle\Theta={\rm Theta}$

$\displaystyle\frac{d\Theta}{d\theta}={\rm Theta'}$

$\displaystyle\frac{d^2\Theta}{d\theta^2}={\rm Theta''}$

$\displaystyle \alpha_{m,n}={\rm alphamn}$

$\displaystyle A_m={\rm Am}$

$\displaystyle B_m={\rm Bm}$

$\displaystyle A_{m,n}={\rm Amn}$

$\displaystyle B_{m,n}={\rm Bmn}$