For the PDE in 29, assume that the following boundary

Chapter 10, Problem 32E

(choose chapter or problem)

For the PDE in Problem 29 , assume that the following boundary conditions are imposed:

$u(0, y, t)=u(a, y, t)=0 ; \quad 0 \leq y \leq b, \quad t \geq 0$

$\frac{\partial u}{\partial y}(x, 0, t)=\frac{\partial u}{\partial y}(x, b, t)=0 ; \quad 0 \leq x \leq a, t \geq 0$

Show that a nontrivial solution of the form $u(x, y, t)=X(x) Y(y) T(t)$ must satisfy the boundary conditions

$X(0)=X(a)=0$,

$Y^{\prime}(0)=Y^{\prime}(b)=0$.

Equation Transcription:

$u(0,y,t)=u(a,y,t)=0;\ \ \ \ \ 0\leq{}y\leq{}b,\ \ \ t\geq{}0$

$\frac{\partial{}u}{\partial{}y}(x,0,t)=\frac{\partial{}u}{\partial{}y}(x,b,t)=0;\ \ \ 0\leq{}x\leq{}a,\ \ t\geq{}0$

$u(x,y,t)=X(x)Y(y)T(t)$

$X(0)=X(a)=0$

$Y'(0)=Y'(b)=0$

Text Transcription:

u(0, y, t)=u(a, y, t)=0 ; \quad 0 \leq y \leq b, \quad t \geq 0

\partial u\partial y(x, 0, t)=\partial u\partial y(x, b, t)=0 ; \quad 0 \leq x \leq a, t \geq 0

u(x,y,t)=X(x)Y(y)T(t)

X(0)=X(a)=0

Y'(0)=Y'(b)=0

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