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The Hermite Equation. The equation y__ 2xy_ + y = 0, < x < , where is a constant, is
Chapter 5, Problem 18(choose chapter or problem)
The Hermite Equation. The equation y__ 2xy_ + y = 0, < x < , where is a constant, is known as the Hermite5 equation. It is an important equation in mathematical physics. a. Find the first four nonzero terms in each of two solutions about x = 0 and show that they form a fundamental set of solutions. b. Observe that if is a nonnegative even integer, then one or the other of the series solutions terminates and becomes a polynomial. Find the polynomial solutions for = 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 = 2n for which the coefficient of xn is 2n. Find H0( x), H1( x), . . . , H5( x).
Questions & Answers
QUESTION:
The Hermite Equation. The equation y__ 2xy_ + y = 0, < x < , where is a constant, is known as the Hermite5 equation. It is an important equation in mathematical physics. a. Find the first four nonzero terms in each of two solutions about x = 0 and show that they form a fundamental set of solutions. b. Observe that if is a nonnegative even integer, then one or the other of the series solutions terminates and becomes a polynomial. Find the polynomial solutions for = 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 = 2n for which the coefficient of xn is 2n. Find H0( x), H1( x), . . . , H5( x).
ANSWER:Step 1 of 24
Consider the Hermite differential equation
Here, 
(a) The objective is to find the first four terms in each of two solutions about x=0 and show that they form a fundamental set of solutions.