91Ó°ÊÓ

Find (a) the Fourier sine series of \(f\) on \(0 \leq x \leq \pi\), and \((b)\) the Fourier cosine series of \(f\) on \(0 \leq x \leq \pi\). $$ f(x)=1,0 \leq x \leq \pi $$

$$ \frac{d^{2} y}{d x^{2}}+\lambda y=0, \quad y(0)=0, \quad y(\pi / 2)=0 . $$

Consider the set of functions \(\left\\{\phi_{n}\right\\}\), where $$ \begin{aligned} \phi_{1}(x) &=\frac{1}{\sqrt{\pi}} \\ \phi_{n+1}(x) &=\sqrt{\frac{2}{\pi}} \cos n x \quad(n=1,2,3, \ldots) \end{aligned} $$ on the interval \(0 \leq x \leq \pi\). Show that this set \(\left\\{\phi_{n}\right\\}\) is an orthonormal system with respect to the weight function having the constant value 1 on \(0 \leq x \leq \pi\).

Find the characteristic values and characteristic functions of each of the following Sturm-Liouville problems. $$ \frac{d^{2} y}{d x^{2}}+\lambda y=0, \quad y(0)=0, \quad y\left(\frac{\pi}{2}\right)=0 $$

Obtain the formal expansion of the function \(f\) defined by \(f(x)=x(0 \leq x \leq \pi)\), in a series of orthonormal characteristic functions \(\left\\{\phi_{n}\right\\}\) of the Sturm-Liouville problem $$ \begin{aligned} \frac{d^{2} y}{d x^{2}}+\lambda y &=0 \\ y(0) &=0 \\ y(x) &=0 \end{aligned} $$

Find (a) the Fourier sine series of \(f\) on \(0 \leq x \leq \pi\), and \((b)\) the Fourier cosine series of \(f\) on \(0 \leq x \leq \pi\). $$ f(x)=x, \quad 0 \leq x \leq \pi $$

$$ \frac{d^{2} y}{d x^{2}}+\lambda y=0, \quad y^{\prime}(0)=0, \quad y^{\prime}(L)=0 $$

Find the characteristic values and characteristic functions of each of the following Sturm-Liouville problems. $$ \frac{d^{2} y}{d x^{2}}+\lambda y=0, \quad y(0)=0, \quad y^{\prime}(\pi)=0 . $$

Find the characteristic values and characteristic functions of each of the following Sturm-Liouville problems. $$ \frac{d^{2} y}{d x^{2}}+\lambda y=0, \quad y(0)=0, \quad y(L)=0, \text { where } L>0 . $$

Find (a) the Fourier sine series of \(f\) on \(0 \leq x \leq \pi\), and \((b)\) the Fourier cosine series of \(f\) on \(0 \leq x \leq \pi\). $$ f(x)=\left\\{\begin{array}{ll} 0, & 0 \leq x<\pi / 2 \\ 2, & \pi / 2 \leq x \leq \pi \end{array}\right. $$