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**Chapter 4 / problem involving fourier series **

« **on:**December 11, 2016, 10:31:05 PM »

Problem: Solve by Fourier method $$\begin{align} & u_{tt}-u_{xx}=0\qquad -\frac{\pi}{2}<x<\frac{\pi}{2},\label{1-1}\\ & u_x|_{x=-\pi/2}=u_x|_{x=\pi/2}=0,\label{1-2}\\ &u| _{t=0}=x^2,\qquad u_t|_{t=0}=0.\label{1-3} \end{align}$$

I have $X(x) = A\sin(\sqrt\lambda x) +B\cos(\sqrt\lambda x)$ Using (2), I can either take

$X(x) = A\sin(nx), \lambda = n^2$ OR $X(x) = B\cos(2nx), \lambda = 4n^2$ Is it correct to take either one? Or is there a right one?

I have $X(x) = A\sin(\sqrt\lambda x) +B\cos(\sqrt\lambda x)$ Using (2), I can either take

$X(x) = A\sin(nx), \lambda = n^2$ OR $X(x) = B\cos(2nx), \lambda = 4n^2$ Is it correct to take either one? Or is there a right one?