In 1–5, find a formal solution to the given boundary value

Chapter 10, Problem 2E

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In Problems 1-5, find a formal solution to the given boundary value problem.

$\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0, \quad 0<x<\pi, \quad 0<y<\pi$,

$\frac{\partial u}{\partial x}(0, y)=\frac{\partial u}{\partial x}(\pi, y)=0, \quad 0 \leq y \leq \pi$,

$u(x, 0)=\cos x-2 \cos 4 x, \quad 0 \leq x \leq \pi$,

$u(x, \pi)=0, \quad 0 \leq x \leq \pi$

Equation Transcription:

$\frac{{\partial{}}^{2}u}{{\partial{}x}^{2}}=\frac{{\partial{}}^{2}u}{{\partial{}y}^{2}}=0,\ \ \ 0<x<\pi{},\ \ 0<y<\pi{}$

$\frac{\partial{}u}{\partial{}x}(0,y)=\frac{\partial{}u}{\partial{}x}(\pi{},y)=0,\ \ \ 0\leq{}y\leq{}\pi{}$

$u(x,0)=cos\ x-2\ cos\ 4x,\ \ 0\leq{}x\leq{}\pi{}$

$u(x,\pi{})=0.\ \ 0\leq{}x\leq{}\pi{}$

Text Transcription:

partial^2u/partial x^2=partial^2 u/partial y^2=0, 0<x<pi, 0<y<pi

Partial u/partial x(0,y)=partial u/partial x(pi,y)=0, 0 leq y leq pi

u(x,0)=cos x-2 cos 4x, 0 leq x leq pi

u(x,)=0. 0 leq x leq pi

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