91Ó°ÊÓ

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 \(\lambda_{1}, \lambda_{2}, \lambda_{1}\), and \(\lambda_{4}\). Give the eigenfunctions corresponding to these approximations. $$ y^{\prime \prime}+\lambda y=0, \quad y^{\prime}(0)=0, y(1)+y^{\prime}(1)=0 $$

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

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

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

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

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

Find the Fourier series of \(f\) on the given interval. $$ f(x)=\left\\{\begin{array}{cc} -1, & -\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 \(\lambda_{1}, \lambda_{2}, \lambda_{1}\), and \(\lambda_{4}\). Give the eigenfunctions corresponding to these approximations. $$ y^{\prime \prime}+\lambda y=0, \quad y(0)+y^{\prime}(0)=0, y(1)=0 $$

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. \(f_{1}(x)=e^{x}, f_{2}(x)=x e^{-x}-e^{-x} ; \quad[0,2]\)