1) Define a sequence of polynomials H n (x ) by H 0 (x )=1, H...

1) Define a sequence of polynomials H n (x ) by H 0 (x )=1, H 1 (x )=2 x , and for n>1 by H n+1 (x )=2 x H n (x )−2 n H n−1 (x ) . These polynomials are called Hermite polynomials of degree n. Calculate the first 7 Hermite polynomials of degree less than 7. You can check your results by comparing them to the list of Hermite polynomials on wikipedia (physicist's Hermite polynomials).

2) Use the power series method to solve the differential equation y ' '−2 x y '+λ y=0 where λ is an arbitrary constant. Verify that you get two independent solution y1, y2 by choosing a0=1, a1=0 and a0=0 , a1=1 . Show that the series expansion for one of the two solutions will terminate resulting in a polynomial solution when λ is chosen to be a positive even integer, λ=2 ,4,6,8,10 ,12 ,14 ,.... Rescale the polynomial solution so it starts with 2 n x n + lower powers of x , n=λ/2. Calculate the list of polynomials obtained this way and compare them to your solution of problem 1)

Expert Solution

Colby Messinger answered 2 years ago

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