91Ó°ÊÓ

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

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

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

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

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

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

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

In Problems 7-10 expand the given function in a Fourier-Bessel series using Bessel functions of the same order as in the indicated boundary condition. $$ \begin{aligned} &f(x)=x^{2}, \quad 0

Determine whether the function is even, odd, or neither. $$ f(x)=2|x|-1 $$

Without referring back to the text. Fill in the blank or answer true/false. The set \(\left\\{P_{n}(x)\right\\}\) of Legendre polynomials is orthogonal with respect to the weight function \(p(x)=1\) on \([-1,1]\) and \(P_{0}(x)=1\). Hence \(\int_{-1}^{1} P_{n}(x) d x=\)_____ for \(n>0\).