The equation y'' - 2xy' + ny = 0(adsbygoogle = window.adsbygoogle || []).push({});

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).

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# Hermite's Equation

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