Chapter 12

In Problems 1 and 2, find the eigenfunctions and the equation that defines the eigenvalues for the given boundary-value problem. Use a CAS to approximate the first four eigenvalues \(\lambda_{1}, \lambda_{2}, \lambda_{3}\), and \(\lambda_{4} .\) Give the eigenfunctions corresponding to these approximations. \(y^{\prime \prime}+\lambda y=0, y^{\prime}(0)=0, y(1)+y^{\prime}(1)=0\)

Find the complex Fourier series of \(f\) on the given interval. $$ f(x)=\left\\{\begin{array}{lr} -1, & -2

Show that the given functions are orthogonal on the indicated interval. $$ f_{1}(x)=x, f_{2}(x)=x^{2} ; \quad[-2,2] $$

Determine whether the function is even, odd, or neither. $$ f(x)=\sin 3 x $$

In Problems, find the Fourier series of \(f\) on the given interval. $$ f(x)=\left\\{\begin{array}{lr} 0, & -\pi

Find the eigenfunctions and the equation that defines the eigenvalues for the given boundary-value problem. Use a CAS to approximate the first four eigenvalues \(\lambda_{1}, \lambda_{2}, \lambda_{3}\), and \(\lambda_{4} .\) Give the eigenfunctions corresponding to these approximations. \(y^{\prime \prime}+\lambda y=0, y(0)+y^{\prime}(0)=0, y(1)=0\)

Determine whether the function is even, odd, or neither. $$ f(x)=x \cos x $$

Show that the given functions are orthogonal on the indicated interval. $$ f_{1}(x)=x^{3}, f_{2}(x)=x^{2}+1 ; \quad[-1,1] $$

In Problems, find the Fourier series of \(f\) on the given interval. $$ f(x)=\left\\{\begin{array}{lr} -1, & -\pi

Consider \(y^{\prime \prime}+\lambda y=0\) subject to \(y^{\prime}(0)=0, y^{\prime}(L)=0\). Show that the eigenfunctions are $$ \left\\{1, \cos \frac{\pi}{L} x, \cos \frac{2 \pi}{L} x, \ldots\right\\} $$ This set, which is orthogonal on \([0, L]\), is the basis for the Fourier cosine series.

Find the complex Fourier series of \(f\) on the given interval. $$ f(x)=\left\\{\begin{array}{lr} 0, & -\frac{1}{2}

Determine whether the function is even, odd, or neither. $$ f(x)=x^{2}+x $$

In Problems, find the Fourier series of \(f\) on the given interval. $$ f(x)=\left\\{\begin{array}{lr} 1, & -1

Show that the given functions are orthogonal on the indicated interval. $$ f_{1}(x)=e^{x}, f_{2}(x)=x e^{-x}-e^{-x} ; \quad[0,2] $$

Consider \(y^{\prime \prime}+\lambda y=0\) subject to the periodic boundary conditions \(y(-L)=y(L), y^{\prime}(-L)=y^{\prime}(L)\). Show that the eigenfunctions are \(\left\\{1, \cos \frac{\pi}{L} x, \cos \frac{2 \pi}{L} x, \ldots, \sin \frac{\pi}{L} x, \sin \frac{2 \pi}{L} x, \sin \frac{3 \pi}{L} x, \ldots\right\\}\) This set, which is orthogonal on \([-L, L]\), is the basis for the Fourier series.

Show that the given functions are orthogonal on the indicated interval. $$ f_{1}(x)=\cos x, f_{2}(x)=\sin ^{2} x ; \quad[0, \pi] $$

Determine whether the function is even, odd, or neither. $$ f(x)=x^{3}-4 x $$

In Problems, find the Fourier series of \(f\) on the given interval. $$ f(x)=\left\\{\begin{array}{lr} 0, & -1

Find the complex Fourier series of \(f\) on the given interval. $$ f(x)=x, 0

Show that the given functions are orthogonal on the indicated interval. $$ f_{1}(x)=x, f_{2}(x)=\cos 2 x ; \quad[-\pi / 2, \pi / 2] $$

Determine whether the function is even, odd, or neither. $$ f(x)=e^{x \mid} $$

In Problems, find the Fourier series of \(f\) on the given interval. $$ f(x)=\left\\{\begin{array}{lr} 0, & -\pi

Find the complex Fourier series of \(f\) on the given interval. $$ f(x)=e^{-|x|},-1

Show that the given functions are orthogonal on the indicated interval. $$ f_{1}(x)=e^{x}, f_{2}(x)=\sin x ; \quad[\pi / 4,5 \pi / 4] $$

Determine whether the function is even, odd, or neither. $$ f(x)=e^{x}-e^{-x} $$

In Problems, find the Fourier series of \(f\) on the given interval. $$ f(x)=\left\\{\begin{array}{lr} \pi^{2}, & -\pi

(a) Find the eigenvalues and eigenfunctions of the boundaryvalue problem $$ x^{2} y^{\prime \prime}+x y^{\prime}+\lambda y=0, y(1)=0, y(5)=0 . $$ (b) Put the differential equation in self-adjoint form. (c) Give an orthogonality relation.

Expand the given function in a FourierBessel series using Bessel functions of the same order as in the indicated boundary condition. $$ \begin{aligned} &f(x)=5 x, 0

Show that the given set of functions is orthogonal on the indicated interval. Find the norm of each function in the set. $$ \\{\sin x, \sin 3 x, \sin 5 x, \ldots\\} ; \quad[0, \pi / 2] $$

Determine whether the function is even, odd, or neither. $$ f(x)=\left\\{\begin{array}{lr} x^{2}, & -1

In Problems, find the Fourier series of \(f\) on the given interval. $$ f(x)=x+\pi, \quad-\pi

(a) Find the eigenvalues and eigenfunctions of the boundaryvalue problem $$ y^{\prime \prime}+y^{\prime}+\lambda y=0, y(0)=0, y(2)=0 $$ (b) Put the differential equation in self-adjoint form. (c) Give an orthogonality relation.

Expand the given function in a FourierBessel series using Bessel functions of the same order as in the indicated boundary condition. $$ \begin{aligned} &f(x)=x^{2}, 0

Show that the given set of functions is orthogonal on the indicated interval. Find the norm of each function in the set. $$ \\{\cos x, \cos 3 x, \cos 5 x, \ldots\\} ; \quad[0, \pi / 2] $$

Determine whether the function is even, odd, or neither. $$ f(x)=\left\\{\begin{array}{lr} x+5, & -2

In Problems, find the Fourier series of \(f\) on the given interval. $$ f(x)=3-2 x, \quad-\pi

Show that the given set of functions is orthogonal on the indicated interval. Find the norm of each function in the set. $$ \\{\sin n x\\}, n=1,2,3, \ldots ; \quad[0, \pi] $$

Determine whether the function is even, odd, or neither. $$ f(x)=x^{3}, 0 \leq x \leq 2 $$

In Problems, find the Fourier series of \(f\) on the given interval. $$ f(x)=\left\\{\begin{array}{lr} 0, & -\pi

Show that the given set of functions is orthogonal on the indicated interval. Find the norm of each function in the set. $$ \left\\{\sin \frac{n \pi}{p} x\right\\}, n=1,2,3, \ldots ; \quad[0, p] $$

Sketch the given periodic wave. Find the frequency spectrum of \(f\). $$ f(x)=\left\\{\begin{array}{ll} \cos x, & 0

Determine whether the function is even, odd, or neither. $$ \begin{aligned} &f(x)=\left|x^{5}\right|\\\ &\text { iven function in an ap } \end{aligned} $$

Consider the regular Sturm-Liouville problem: \(\frac{d}{d x}\left[\left(1+x^{2}\right) y^{\prime}\right]+\frac{\lambda}{1+x^{2}} y=0, \quad y(0)=0, \quad y(1)=0\) (a) Find the eigenvalues and eigenfunctions of the boundaryvalue problem. [Hint: Let \(x=\tan \theta\) and then use the Chain Rule.] (b) Give an orthogonality relation.

(a) Use aCASto graph \(y=3 J_{1}(x)+x J_{1}^{\prime}(x)\) onaninterval so that the first five positive \(x\) -intercepts of the graph are shown. (b) Use the root-finding capability of your CAS to approximate the first five roots \(x_{i}\) of the equation $$ 3 J_{1}(x)+x J_{1}^{\prime}(x)=0. $$ (c) Use the data obtained in part (b) to find the first five positive values of \(\alpha_{i}\) that satisfy $$ 3 J_{1}(4 \alpha)+4 \alpha J_{1}^{\prime}(4 \alpha)=0. $$ (d) If instructed, find the first 10 positive values of \(\alpha_{i}\).

Expand the given function in an appropriate cosine or sine series. $$ f(x)=\left\\{\begin{array}{lr} \pi, & -1

Show that the given set of functions is orthogonal on the indicated interval. Find the norm of each function in the set. $$ \begin{aligned} &\left\\{1, \cos \frac{n \pi}{p} x, \sin \frac{m \pi}{p} x\right\\}, n=1,2,3, \ldots, \\ &m=1,2,3, \ldots ; \quad[-p, p] \end{aligned} $$

Expand the given function in an appropriate cosine or sine series. $$ f(x)=\left\\{\begin{array}{lr} 1, & -2

In Problems, find the Fourier series of \(f\) on the given interval. $$ f(x)=\left\\{\begin{array}{lr} 0, & -2

Verify by direct integration that the functions are orthogonal with respect to the indicated weight function on the given interval. $$ \begin{aligned} &H_{0}(x)=1, H_{1}(x)=2 x, H_{2}(x)=4 x^{2}-2 ; \quad w(x)=e^{-x^{2}}, \\ &(-\infty, \infty) \end{aligned} $$

In Problems, find the Fourier series of \(f\) on the given interval. $$ f(x)=\left\\{\begin{array}{lr} 1, & -5