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Chapter 6: Problem 27

Find the eigenvalues and eigenfunctions of the given Sturm-Liouville lems: a. \(y^{\prime \prime}+\lambda y=0, y^{\prime}(0)=0=y^{\prime}(\pi)\) b. \(\left(x y^{\prime}\right)^{\prime}+\frac{\lambda}{x} y=0, y(1)=y\left(e^{2}\right)=0\).

Short Answer

Expert verified
The eigenvalues and eigenfunctions are solved by transforming the given problem into a standard Sturm-Liouville problem and then solving that problem, for each of the two Sturm-Liouville problems given.

Step by step solution

01

Solve for eigenvalues and eigenfunctions of first problem

For the first Sturm-Liouville problem, \(y^{\prime \prime}+\lambda y=0, y^{\prime}(0)=0=y^{\prime}(\pi)\), the aim is to find \(\lambda\) such that nontrivial solutions exist. Try the solutions of the form \(y=Acos(\sqrt{\lambda}x) + Bsin(\sqrt{\lambda}x)\) where A and B are coefficients to be determined. Substituting this into the given equation and applying the given boundary conditions will give the values of A, B and \(\lambda\).
02

Solve for eigenvalues and eigenfunctions of the second problem

For the second problem given, \(\left(x y^{\prime}\right)^{\prime}+\frac{\lambda}{x} y=0, y(1)=y\left(e^{2}\right)=0\), approach it similarly. Try the solutions of the form \(y=x^{m}\). Substituting this into the equation and applying the boundary conditions will give the values of m and \(\lambda\). Solving for \(\lambda\) will get the eigenvalues and keeping those \(\lambda\) values into the equation will yield the corresponding eigenfunctions.

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Most popular questions from this chapter

Consider the set of vectors \((-1,1,1),(1,-1,1),(1,1,-1)\) a. Use the Gram-Schmidt process to find an orthonormal basis for \(R^{3}\) using this set in the given order. b. What do you get if you do reverse the order of these vectors?

Bessel functions \(J_{p}(\lambda x)\) are solutions of \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(\lambda^{2} x^{2}-p^{2}\right) y=\) 0. Assume that \(x \in(0,1)\) and that \(J_{p}(\lambda)=0\) and \(J_{p}(0)\) is finite.s a. Show that this equation can be written in the form $$ \frac{d}{d x}\left(x \frac{d y}{d x}\right)+\left(\lambda^{2} x-\frac{p^{2}}{x}\right) y=0 $$ This is the standard Sturm-Liouville form for Bessel's equation. b. Prove that $$ \int_{0}^{1} x J_{p}(\lambda x) J_{p}(\mu x) d x=0, \quad \lambda \neq \mu $$ by considering $$ \int_{0}^{1}\left[J_{p}(\mu x) \frac{d}{d x}\left(x \frac{d}{d x} J_{p}(\lambda x)\right)-J_{p}(\lambda x) \frac{d}{d x}\left(x \frac{d}{d x} J_{p}(\mu x)\right)\right] d x $$ Thus, the solutions corresponding to different eigenvalues \((\lambda, \mu)\) are orthogonal. c. Prove that $$ \int_{0}^{1} x\left[J_{p}(\lambda x)\right]^{2} d x=\frac{1}{2} J_{p+1}^{2}(\lambda)=\frac{1}{2} J_{p}^{\prime 2}(\lambda) $$

Prove that if \(u(x)\) and \(v(x)\) satisfy the general homogeneous boundary itions $$ \begin{aligned} &\alpha_{1} u(a)+\beta_{1} u^{\prime}(a)=0 \\ &\alpha_{2} u(b)+\beta_{2} u^{\prime}(b)=0 \end{aligned} $$ \(=a\) and \(x=b\), then $$ p(x)\left[u(x) v^{\prime}(x)-v(x) u^{\prime}(x)\right]_{x=a}^{x=b}=0 $$

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 eral Sturm-Liouville operator \(\mathcal{L}\).

Expand the following in a Fourier-Legendre series for \(x \in(-1,1)\). a. \(f(x)=x^{2}\). b. \(f(x)=5 x^{4}+2 x^{3}-x+3\). c. \(f(x)=\left\\{\begin{array}{cc}-1, & -1
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