Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 6: Problem 14

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 \(j=2 k\) : $$ J_{n}(x)=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{k !(n+k) !}\left(\frac{x}{2}\right)^{n+2 k} $$

Short Answer

Expert verified
We successfully demonstrated that with the initial condition for \(a_0\) given and applying the recurrence relation, the function \(y(x)\) does represent the Bessel function of the first kind of order \(n\), precisely \(J_{n}(x)=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{k !(n+k)!}\left(\frac{x}{2}\right)^{n+2 k}\).

Step by step solution

01

Understanding the initial given condition

According to the exercise, a power series solution of the form \(y(x)=\sum_{j=0}^{\infty} a_{j} x^{j+n}\) is proposed, where \(a_j\) is a recurrence relation. This leads to the power series representation of \(y(x)\), which goes as \(a_{0} x^n, a_{1} x^{n+1}, a_{2} x^{n+2}, \ldots\). Additionally, \(a_{j}=\frac{-1}{j(2 n+j)} a_{j-2}\) is known.
02

Determining the value of the coefficients for even j values

In order to find the coefficients with even indices \(j=2k\), where \(k\) is an integer, we can iterate \(a_{j}=\frac{-1}{j(2 n+j)} a_{j-2}\), starting from \(j=0\). We get the following:\n- \(a_{2}= \frac{-1}{2(2 n+2)} a_{0}\),\n- \(a_{4}= \frac{-1}{4(2 n+4)} a_{2} = \frac{-1}{4(2 n+4)} \cdot \frac{-1}{2(2 n+2)} a_{0}\), and so on.\nGenerally, as a result, we find that for \(j = 2 k\), \(a_{2k}= \frac{(-1)^k}{2^k k! (2 n+2)(2 n+4) \ldots (2 n + 2k)} a_{0}\).
03

Substituting a0 value and simplifying the expression for a2k

Substituting the given \(a_{0}\) value into the equation for \(a_{2k}\), we get: \(a_{2k}= \frac{(-1)^k}{2^k k!(n+1) \ldots (n+k)} \left(\frac{x}{2}\right)^{n+2k}\).
04

Expressing y(x) as a power series with a2k coefficients

Substituting these values back into the expression of \(y(x)\), we get: \(y(x)=\sum_{k=0}^{\infty}\frac{(-1)^k}{k!(n+1)\ldots(n+k)} \left(\frac{x}{2}\right)^{n+2k}\). Notice that the term in the denominator \(k!(n+1) \ldots (n+k) = (n+k)!\). Thus we reach the definition of \(J_n(x)\): \(J_{n}(x)=\sum_{k=0}^{\infty} \frac{(-1)^k}{k!(n+k)!} \left(\frac{x}{2}\right)^{n+2k}\).
05

Comparing the resulting expression with the Bessel function

Comparing our result with the Bessel function defined in the problem, it's clear that these expressions coincide. Thus, we have demonstrated that the initial y(x) expression indeed represents the Bessel function of the first kind of order n.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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

Consider the boundary value problem: \(y^{\prime \prime}-y=x, x \in(0,1)\), with Idary conditions \(y(0)=y(1)=0\). a. Find a closed form solution without using Green's functions. b. Determine the closed form Green's function using the properties of Green's functions. Use this Green's function to obtain a solution of the boundary value problem. c. Determine a series representation of the Green's function. Use this Green's function to obtain a solution of the boundary value problem. d. Confirm that all of the solutions obtained give the same results.

The coefficients \(C_{k}^{p}\) in the binomial expansion for \((1+x)^{p}\) are given by $$ C_{k}^{p}=\frac{p(p-1) \cdots(p-k+1)}{k !} $$ a. Write \(C_{k}^{p}\) in terms of Gamma functions. b. For \(p=1 / 2\), use the properties of Gamma functions to write \(C_{k}^{1 / 2}\) in terms of factorials. c. Confirm your answer in part b. by deriving the Maclaurin series expansion of \((1+x)^{1 / 2}\)

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

Use the method of eigenfunction expansions to solve the problems: a. \(y^{\prime \prime}+4 y=x^{2}, \quad y^{\prime}(0)=y^{\prime}(1)=0\). b. \(y+4 y=x^{2} \quad y(0)=y(1)=0\).
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free