91Ó°ÊÓ

In Exercises \(1-8\), verify that the given function is biharmonic. Use the result of Example 1 where appropriate. In what follows, \((x, y)\) denote Cartesien coordinates, \((r, \theta)\) polar coordinates, and \(n\) an integer. \(r^{n+2} \cos n \theta\)

In exercises 1-10, put the given equation in Sturm-Liouville form and decide whether, the problem is regular or singular. $$ y^{\prime \prime}+\lambda x y=0, y(-1)=0, y(1)=0 $$

In Exercises \(1-8\), verify that the given function is biharmonic. Use the result of Example 1 where appropriate. In what follows, \((x, y)\) denote Cartesien coordinates, \((r, \theta)\) polar coordinates, and \(n\) an integer. \(r^{2} \ln r\)

In exercises 1-10, put the given equation in Sturm-Liouville form and decide whether, the problem is regular or singular. $$ \left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+(1+\lambda x) y=0, y(-1)=0, y(1)=0 $$

In exercises 1-10, put the given equation in Sturm-Liouville form and decide whether, the problem is regular or singular. $$ \left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+\lambda y=0, y(-1)=0, y(1)=0 $$

Determine the constants \(a\) and \(b\) so that the functions \(1, x\), and \(a+b x+x^{2}\) become orthogonal on the interval \([-1,1]\).

(a) Show that the eigenvalues \(\lambda\) of $$ \begin{gathered} X^{(4)}-\lambda X=0 \\ X^{\prime \prime}(0)=0, X^{\prime \prime}(0)=0, X^{\prime \prime}(L)=0, X^{\prime \prime \prime}(L)=0 \end{gathered} $$ are \(\lambda_{1}=0, \lambda_{2}=0\), and all values of the form \(\lambda=\alpha^{4}\), where \(\alpha\) is a positive toot of cosh \(\alpha L \cos \alpha L=1\). Note that, aside from the eigenvalue 0 , these are procisely the, eigenvalues of Example 1. (b) Derive the eigenfunctions \(X_{1}=1, \quad X_{2}=x-\frac{L}{2}\) for \(n=3,4, \ldots\), where \(\alpha_{n+2}\) is the nth positive root of \(\cosh \alpha L \cos \alpha L=1\).

In Frercises 11-20, determine the eigenvalues and eigenfunctions of the given SturmLiosville problem. $$ y^{\prime \prime}+\lambda y=0, y(0)=0, y(2 \pi)=0 $$

In Frercises 11-20, determine the eigenvalues and eigenfunctions of the given SturmLiosville problem. $$ y^{\prime \prime}+\lambda y=0, y(-\pi)=y(\pi), y^{\prime}(-\pi)=y^{\prime}(\pi) $$

What is the orthonormal set corresponding to the Legendre polynomials on the interval \([-1,1 \mid ?\)