In 1?4, find a formal solution to the vibrating string

Chapter 10, Problem 3E

(choose chapter or problem)

In Problems $1-4$ , find a formal solution to the vibrating string problem governed by the given initial-boundary value problem.

$\frac{\partial^{2} u}{\partial t^{2}}=4 \frac{\partial^{2} u}{\partial x^{2}}, \quad 0<x<\pi, \quad t>0$

$u(0, t)=u(\pi, t)=0, t>0$

$u(x, 0)=x^{2}(\pi-x), 0<x<\pi$

$\frac{\partial u}{\partial t}(x, 0)=0, \quad 0<x<\pi$

Equation Transcription:

$\frac{{\partial{}}^{2}u}{\partial{}{t}^{2}}=4\frac{{\partial{}}^{2}u}{\partial{}{x}^{2}},\ \ \ 0<x<\pi{},\ \ t>0$

$u(0,t)=u(\pi{},t)=0,\ t>0$

$u(x,0)={x}^{2}(\pi{}-x),\ \ 0<x<\pi{}$

$\frac{\partial{}u}{\partial{}t}(x,0)=0,\ \ 0<x<\pi{}$

Text Transcription:

\partial^2 u\partial t^2=4 \partial^2 u\partial x^2, \quad 0<x<\pi, \quad t>0

u(0, t)=u(\pi, t)=0, t>0

u(x, 0)=x^2(\pi-x), 0<x<\pi

\partial u \partial t}(x, 0)=0, \quad 0<x<\pi

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