Exploring Hermite's Equation: Solutions, Polynomials & Coefficients

[nintandao64]

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!

 

[vela]

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!

You need to show some work before getting help here. The problem wants you to find a series solution to the differential equation, so start with

$$y=a_0+a_1 x+a_2 x^2+a_3 x^3 + \cdots$$

plug it into the differential equation, and solve for the a_n's.