6.5: Problems

6.1. Find the adjoint operator and its domain for \(L u=u^{\prime \prime}+4 u^{\prime}-3 u, u^{\prime}(0)+ 4 u(0)=0, u^{\prime}(1)+4 u(1)=0\)

6.2. Show that a Sturm-Liouville operator with periodic boundary conditions on \([a, b]\) is self-adjoint if and only if \(p(a)=p(b)\). [Recall, periodic boundary conditions are given as \(u(a)=u(b)\) and \(u^{\prime}(a)=u^{\prime}(b)\).]

6.3. The Hermite differential equation is given by \(y^{\prime \prime}-2 x y^{\prime}+\lambda y=0\). Rewrite this equation in self-adjoint form. From the Sturm-Liouville form obtained, verify that the differential operator is self adjoint on \((-\infty, \infty)\). Give the integral form for the orthogonality of the eigenfunctions.

6.4. Find the eigenvalues and eigenfunctions of the given Sturm-Liouville problems.

a. \(y^{\prime \prime}+\lambda y=0, y^{\prime}(0)=0=y^{\prime}(\pi)\).
b. \(\left(x y^{\prime}\right)^{\prime}+\dfrac{\lambda}{x} y=0, y(1)=y\left(e^{2}\right)=0\).

6.5. The eigenvalue problem \(x^{2} y^{\prime \prime}-\lambda x y^{\prime}+\lambda y=0\) with \(y(1)=y(2)=0\) is not a Sturm-Liouville eigenvalue problem. Show that none of the eigenvalues are real by solving this eigenvalue problem.

6.6. In Example 6.10 we found a bound on the lowest eigenvalue for the given eigenvalue problem.

a. Verify the computation in the example.

b. Apply the method using

\(y(x)=\left\{\begin{array}{cc}
x, & 0<x<\dfrac{1}{2} \\
1-x, & \dfrac{1}{2}<x<1
\end{array}\right.\)

Is this an upper bound on \(\lambda_{1}\)

c. Use the Rayleigh quotient to obtain a good upper bound for the lowest eigenvalue of the eigenvalue problem: \(\phi^{\prime \prime}+\left(\lambda-x^{2}\right) \phi=0, \phi(0)=0\), \(\phi^{\prime}(1)=0\)

6.7. Use the method of eigenfunction expansions to solve the problem:

\[y^{\prime \prime}+4 y=x^{2}, \quad y(0)=y(1)=0 \nonumber \]

6.8. Determine the solvability conditions for the nonhomogeneous boundary value problem: \(u^{\prime \prime}+4 u=f(x), u(0)=\alpha, u^{\prime}(1)=\beta\)