   # Hermite’s Equation Consider Hermite’s Equation y ' ' − 2 x y ' + 2 k y = 0 Use a power series of the form y = ∑ n = 0 ∞ a n ( x ) n to find the solution when k = 4 . [Hint: Choose the arbitrary constants such that the leading term is ( 2 x ) k .] Polynomial solutions of Hermite’s equation are called Hermite polynomials and are denoted by H k ( x ) . The general form of H k ( x ) can be written as H k ( x ) = ∑ n = 0 P ( − 1 ) n k ! ( 2 x ) k − 2 n n ! ( k − 2 n ) ! ### 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 19PS
Textbook Problem
3 views

## Hermite’s Equation Consider Hermite’s Equation y ' ' − 2 x y ' + 2 k y = 0 Use a power series of the form y = ∑ n = 0 ∞ a n ( x ) n to find the solution when k = 4 . [Hint: Choose the arbitrary constants such that the leading term is ( 2 x ) k .]Polynomial solutions of Hermite’s equation are called Hermite polynomials and are denoted by H k ( x ) . The general form of H k ( x ) can be written as H k ( x ) = ∑ n = 0 P ( − 1 ) n k ! ( 2 x ) k − 2 n n ! ( k − 2 n ) !

(a)

To determine

To calculate: The solution of the Hermite’s equation, y''2xy'+2ky=0 when k=4, using a power series.

### Explanation of Solution

Given information:

The given differential equation is y''2xy'+2ky=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=0anxn is a solution of the differential equation. Then, differentiating the assumed solution with respect to x,

y'=n=1nanxn1

Again differentiating above equation with respect to x,

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

Now, the given differential equation is,

y''2xy'+2ky=0

For obtaining the solution when k=4, substitute k=4 in above equation as,

y''2xy'+2(4)y=0y''2xy'+8y=0

Insert the values of y and y'' in above equation,

y''2xy+8y=0n=2n(n1)anxn22xn=1nanxn1+8n=0anxn=0n=2n(n1)anxn22n=0nanxn+8n=0anxn=0n=2n(n1)anxn2=n=0(2n8)anxnn=0(n+2)(n+1)an+2xn=n=0(2n8)anxn

Equating the coefficients of like terms, one can obtain the recursion formula

(b)

To determine

The values of Hermite polynomials H0(x), H1(x), H2(x), H3(x), and H4(x) if Hk(x) is given by Hk(x)=n=0P(1)nk!(2x)k2nn!(k2n)!.

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Find more solutions based on key concepts
Evaluate the integral. 21. dt(t21)2

Single Variable Calculus: Early Transcendentals

In Exercises 85-88, determine whether the statement is true or false. If it is true, explain why it is true. If...

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Solve each equation for x. 51. (a) e74x=6 (b) ln(3x 10) = 2

Single Variable Calculus: Early Transcendentals, Volume I

For f(x) = (x3 + 1) (x4 + 1), f(x) = a) (3x2)(4x3) b) (x3 + 1)(x4) + (x4 + 1)(x3) c) (x3 + 1)(4x3) + (x4 + 1)(3...

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

Graph each piecewise-defined function. fx=xifx<0xifx0

College Algebra (MindTap Course List)

In problems 11-16, write in radical form.Do not simplify. 16.

Mathematical Applications for the Management, Life, and Social Sciences

If 270A360, then is cos(A/2) positive or negative?

Trigonometry (MindTap Course List)

In Problems 3136 use the substitution x = et to transform the given Cauchy-Euler equation to a differential equ...

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

The general fund budget for the state of Kentucky for 1988 (Period 1) to 2011 (Period 24) follows (Northern Ken...

Modern Business Statistics with Microsoft Office Excel (with XLSTAT Education Edition Printed Access Card) (MindTap Course List)

Reminder Round all answers to two decimal places unless otherwise indicated. Using the Laws of LogarithmsFor Ex...

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List) 