   # Bessel’s Equation: Order Zero The differential equation x 2 y ' ' + x y ′ + x 2 y = 0 is known as Bessel’s equation of order zero. Use a power series of the form y = ∑ n = 0 ∞ a n ( x ) n to find the solution. Compare your result with that of the function J 0 ( x ) given in Section 9.8, Exercise 65. ### Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378

#### Solutions

Chapter
Section ### Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378
Chapter 16, Problem 17PS
Textbook Problem
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## Bessel’s Equation: Order Zero The differential equation x 2 y ' ' + x y ′ + x 2 y = 0 is known as Bessel’s equation of order zero.Use a power series of the form y = ∑ n = 0 ∞ a n ( x ) n to find the solution.Compare your result with that of the function J 0 ( x ) given in Section 9.8, Exercise 65.

(a)

To determine

To calculate: The solution of the Bessel’s equation x2y''+xy+x2y=0, using a power series.

### Explanation of Solution

Given information:

The given differential equation is x2y''+xy+x2y=0.

Concept Used:

Power series is defined as an infinite series of the form,

n=0anxn=a0+a1x+a2x2+a3x3+.........+anxn+...., where x is a variable.

Calculation:

Assume y=n=0an(x)n is a solution of the differential equation.

Let the general form of the Bessel’s equation of n order is,

x2y+xy+(x2n2)y=0

On comparing the above equation, we get n=0, as x>0 is given.

Therefore, the order of the equation is zero.

Assume y=n=0anxn is a solution of the differential equation. Then, differentiating the assumed solution with respect to x,

y'=n=1nanxn1y=n=2n(n1)anxn2

Now, the given Bessel’s equation is,

x2y+xy+(x2n2)y=0

Inserting the value of y and y in above equation,

x2n=2n(n1)anxn2+xn=1nanxn1+(x2n2)n=0anxn=0n=2n(n1)anxn+n=1nanxn+(x2n2)n=0anxn=0

The solution of the general equation x2y+xy+(x2n2)y=0 is

y=a0k=0(1)kx2k+n22kk!(1+n)(2+n)

(b)

To determine

Compare the resultobtained from the function J0(x) given in the section 9.8.

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