# 4.5: Problems

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## Exercise $$\PageIndex{1}$$

Prove the if $$u(x)$$ and $$v(x)$$ satisfy the general homogeneous boundary conditions $\begin{array}{r} \alpha_{1} u(a)+\beta_{1} u^{\prime}(a)=0, \\ \alpha_{2} u(b)+\beta_{2} u^{\prime}(b)=0 \end{array}\label{eq:1}$ at $$x=a$$ and $$x=b$$, then $p(x)\left[u(x) v^{\prime}(x)-v(x) u^{\prime}(x)\right]_{x=a}^{x=b}=0 .\nonumber$

## Exercise $$\PageIndex{2}$$

Prove Green’s Identity $$\int_{a}^{b}(u \mathcal{L} v-v \mathcal{L} u) d x=\left.\left[p\left(u v^{\prime}-v u^{\prime}\right)\right]\right|_{a} ^{b}$$ for the general Sturm-Liouville operator $$\mathcal{L}$$.

## Exercise $$\PageIndex{3}$$

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

## Exercise $$\PageIndex{4}$$

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 $$u^{\prime}(a)=u^{\prime}(b)$$.]

## Exercise $$\PageIndex{5}$$

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.

## Exercise $$\PageIndex{6}$$

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

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

## Exercise $$\PageIndex{7}$$

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.

## Exercise $$\PageIndex{8}$$

In Example 4.2.7 we found a bound on the lowest eigenvalue for the given eigenvalue problem.

1. Verify the computation in the example.
2. Apply the method using $y(x)=\left\{\begin{array}{cc} x, & 0<x<\frac{1}{2} \\ 1-x, & \frac{1}{2}<x<1 \end{array}\right.\nonumber$ Is this an upper bound on $$\lambda_{1}$$
3. Use the Rayleigh quotient to obtain a good upper bound for the lowest eigenvalue of the eigenvalue problem: $$\phi^{\prime \prime}+\left(\lambda-x^{2}\right) \phi=0$$, $$\phi(0)=0, \phi^{\prime}(1)=0$$.

## Exercise $$\PageIndex{9}$$

Use the method of eigenfunction expansions to solve the problems:

1. $$y^{\prime \prime}=x^{2}, \quad y(0)=y(1)=0$$.
2. $$y^{\prime \prime}+4 y=x^{2}, \quad y^{\prime}(0)=y^{\prime}(1)=0$$.

## Exercise $$\PageIndex{10}$$

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

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This page titled 4.5: Problems is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Russell Herman via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.