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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

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 $$ p(x)\left[u(x) v^{\prime}(x)-v(x) u^{\prime}(x)\right]_{x=a}^{x=b}=0 $$

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?

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

Determine the solvability conditions for the nonhomogeneous boundralue problem: \(u^{\prime}(\pi / 4)=\beta\).

A solution of Bessel's equation, \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-n^{2}\right) y=0\), can be found using the guess \(y(x)=\sum_{j=0}^{\infty} a_{j} x^{j+n} .\) One obtains the recurrence relation \(a_{j}=\frac{-1}{j(2 n+j)} a_{j-2} .\) Show that for \(a_{0}=\left(n ! 2^{n}\right)^{-1}\), we get the Bessel function of the first kind of order \(n\) from the even values \(j=2 k\) : $$ J_{n}(x)=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{k !(n+k) !}\left(\frac{x}{2}\right)^{n+2 k} $$
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