91Ó°ÊÓ

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=1 ; \quad f(x)=\left\\{\begin{array}{cc} 0, & -1

Solve the eigenvalue problem. $$ y^{\prime \prime}+\lambda y=0, \quad y(0)=0, \quad y(2)=0 $$

Find the Fourier sine series. $$ f(x)=\cos k x(k \neq \text { integer }) ; \quad[0, \pi] $$

Solve the eigenvalue problem. $$ y^{\prime \prime}+\lambda y=0, \quad y^{\prime}(0)=0, \quad y(3)=0 $$

Find the Fourier sine series. $$ f(x)=\left\\{\begin{array}{ll} 1, & 0 \leq x \leq \frac{L}{2} \\ 0, & \frac{L}{2}

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=1 ; \quad f(x)=\left\\{\begin{array}{cc} 0, & -1

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=4 ; \quad f(x)=\left\\{\begin{array}{lr} 0, & -4

Find the Fourier sine series. $$ f(x)=\left\\{\begin{array}{cl} x, & 0 \leq x \leq \frac{L}{2} \\ L-x, & \frac{L}{2} \leq x \leq L \end{array} \quad[0, L]\right. $$

Solve the eigenvalue problem. $$ y^{\prime \prime}+\lambda y=0, \quad y(0)=0, \quad y^{\prime}(1 / 2)=0 $$

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=1 ; \quad f(x)=\left\\{\begin{array}{cr} x^{2}, & -1