How Do You Solve Hermite's Equation?

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SUMMARY

The discussion focuses on solving Hermite's equation, represented as y'' - 2xy' + ny = 0, where n is a constant. Participants are tasked with finding the first four terms of two solutions around x=0, demonstrating that they form a fundamental set of solutions. Additionally, it is noted that for nonnegative even integers n, one series solution terminates, yielding polynomial solutions for n values of 0, 2, 4, 6, 8, and 10. The Hermite polynomial Hn(x) is defined as the polynomial solution with n=2n, specifically requiring the coefficient of x^n to be 2^n.

PREREQUISITES
  • Understanding of differential equations, specifically second-order linear equations.
  • Familiarity with series solutions and power series expansions.
  • Knowledge of polynomial functions and their properties.
  • Basic concepts of orthogonal polynomials, particularly Hermite polynomials.
NEXT STEPS
  • Study the derivation of Hermite polynomials Hn(x) for various values of n.
  • Explore the application of Hermite's equation in quantum mechanics and probability theory.
  • Learn about the properties of orthogonal polynomials and their applications in numerical analysis.
  • Investigate the relationship between Hermite polynomials and other special functions, such as Laguerre and Legendre polynomials.
USEFUL FOR

Mathematicians, physicists, and engineering students who are solving differential equations, particularly those interested in special functions and their applications in various fields.

nintandao64
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The equation y'' - 2xy' + ny = 0
where n is a constant, is known as Hermite's equation
a) Find the first four terms in each of two solutions about x=0 and show that htey form a fundamental set of solutions
b) Observe that if n is a nonnegative even integer, then one or the other of the series solutions terminates and becomes a polynomial. Find the polynomial solutoins for n=0, 2 ,4, 6, 8 and 10. Note that each polynomial is determined only up to a multiplicative constant
c) The Hermite polynomial Hn(x) is defined as the polynomial solution of the Hermite equation with n=2n for which the coefficient of x^n is 2^n. Find H0(x),...,H5(x).

Help!
 
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nintandao64 said:
The equation y'' - 2xy' + ny = 0
where n is a constant, is known as Hermite's equation
a) Find the first four terms in each of two solutions about x=0 and show that htey form a fundamental set of solutions
b) Observe that if n is a nonnegative even integer, then one or the other of the series solutions terminates and becomes a polynomial. Find the polynomial solutoins for n=0, 2 ,4, 6, 8 and 10. Note that each polynomial is determined only up to a multiplicative constant
c) The Hermite polynomial Hn(x) is defined as the polynomial solution of the Hermite equation with n=2n for which the coefficient of x^n is 2^n. Find H0(x),...,H5(x).

Help!

Is this a homework problem? It sure looks like one!

jason
 

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