91Ó°ÊÓ

Solve the eigenvalue problem. $$ y^{\prime \prime}+\lambda y=0, \quad y^{\prime}(0)=0, \quad y(\pi)=0 $$

Solve the eigenvalue problem. $$ y^{\prime \prime}+\lambda y=0, \quad y(-\pi)=y(\pi), \quad y^{\prime}(-\pi)=y^{\prime}(\pi) $$

Find the Fourier cosine series. $$ f(x)=x^{2}-L^{2} ; \quad[0, L] $$

Find the Fourier series of \(f\) on \([-L, L]\) and determine its sum for \(-L \leq x \leq L .\) Where indicated by \(C\), graph \(f\) and $$ F_{m}(x)=a_{0}+\sum_{n=1}^{m}\left(a_{n} \cos \frac{n \pi x}{L}+b_{n} \sin \frac{n \pi x}{L}\right) $$ on the same axes for various values of \(m\). $$ L=\pi ; \quad f(x)=|x| \cos x $$

Find the Fourier cosine series. $$ f(x)=(x-1)^{2} ; \quad[0,1] $$

Solve the eigenvalue problem. $$ y^{\prime \prime}+\lambda y=0, \quad y^{\prime}(0)=0, \quad y^{\prime}(1)=0 $$

Find the Fourier cosine series. $$ f(x)=e^{x} ; \quad[0, \pi] $$

Solve the eigenvalue problem. $$ y^{\prime \prime}+\lambda y=0, \quad y^{\prime}(0)=0, \quad y(1)=0 $$

Find the Fourier series of \(f\) on \([-L, L]\) and determine its sum for \(-L \leq x \leq L .\) Where indicated by \(C\), graph \(f\) and $$ F_{m}(x)=a_{0}+\sum_{n=1}^{m}\left(a_{n} \cos \frac{n \pi x}{L}+b_{n} \sin \frac{n \pi x}{L}\right) $$ on the same axes for various values of \(m\). $$ L=\pi ; \quad f(x)=x \sin x $$

Find the Fourier series of \(f\) on \([-L, L]\) and determine its sum for \(-L \leq x \leq L .\) Where indicated by \(C\), graph \(f\) and $$ F_{m}(x)=a_{0}+\sum_{n=1}^{m}\left(a_{n} \cos \frac{n \pi x}{L}+b_{n} \sin \frac{n \pi x}{L}\right) $$ on the same axes for various values of \(m\). $$ L=\pi ; \quad f(x)=|x| \sin x $$