Note: Use the prime notation for derivatives, so the derivative of $X$ is written as $X^\prime$. Do NOT use $X^\prime(x)$. The variable $\lambda$ is typed lambda, $\mu$ is typed as mu.

Solve the two-dimensional wave equation with clamped edges and the given initial conditions:

Part 1 Separating the variables
We try for a separable solution $u(x,y) = X(x)Y(y)T(t)$, plugging $XYT$ into the PDE for $u$ we get

$=$

Therefore $X^{\prime\prime}YT$ =
Divide both sides by $XYT$ leads to the separated equations:
$=$ $=-\lambda$

For now we will work with the middle of this equation, the equation in Y and T. This equation also separates:
$\displaystyle\frac{Y^{\prime\prime}}{Y} =$ = $-\mu$

We now have three separated differential equations on the three variables:
ODE in X: $=0$ (use m for the multiple of $\pi$)
ODE in Y: $=0$ (use n for the multiple of $\pi$)
ODE in T: $=0$

Part 2 The Sturm-Liouville Problems
From the boundary conditions on the plate we see that
X(0) =
X(b) =
Y(0) =
Y(c) =

That makes the differential equations in X and Y Sturm-Liouville problems. We can use them to find $\lambda$ and $\mu$, and then X,Y, T:
X(x) = (use m as the multiple of $\pi$)
Y(y) = (use n as the multiple of $\pi$)
T(t) = (use Amn as the coefficient on cos, Bmn as the coefficient on sin)

Part 3 The series solution
$u(x,y,t) = \sum\limits_{m=1}^\infty\sum\limits_{n=1}^\infty XYT = \sum\limits_{m=1}^\infty\sum\limits_{n=1}^\infty$

We now need to find the coefficients $A_{m,n}$ and $B_{m,n}$. Using the first initial condition:

$u(x,y,0) = \sum\limits_{m=1}^\infty\sum\limits_{n=1}^\infty$ $\displaystyle = f(x,y)$
Therefore
 $A_{m,n} =$ ---------- $\displaystyle\int$$0$ $\displaystyle\int$$0$ $f(x,y)$ dydx

Now we use the second initial condition:
$\frac{\partial u}{\partial t}\Bigg\vert_{t=0} = \sum\limits_{m=1}^\infty\sum\limits_{n=1}^\infty$ $\displaystyle = g(x,y)$
Therefore
 $B_{m,n} =$ $\hskip 31pt$ ----------------------------------- $\displaystyle\int$$0$ $\displaystyle\int$$0$ $g(x,y)$ dydx

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