# 2 dx dx b show that if 2n 1 where n is a nonnegative

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Unformatted text preview: d a nonzero polynomial solution to this diﬀerential equation, in the case where p = 3. d. Find a basis for the space of solutions to the diﬀerential equation (1 − x2 ) d2 y dy + 12y = 0. − 2x 2 dx dx 1.2.3. The diﬀerential equation (1 − x2 ) d2 y dy −x + p2 y = 0, dx2 dx where p is a constant, is known as Chebyshev’s equation . It can be rewritten in the form d2 y dy + Q(x)y = 0, + P (x) 2 dx dx where 14 P (x) = − x , 1 − x2 Q(x) = p2 . 1 − x2 a. If P (x) and Q(x) are represented as power series about x0 = 0, what is the radius of convergence of these power series? b. Assuming a power series centered at 0, ﬁnd the recursion formula for an+2 in terms of an . c. Use the recursion formula to determine an in terms of a0 and a1 , for 2 ≤ n ≤ 9. d. In the special case where p = 3, ﬁnd a nonzero polynomial solution to this diﬀerential equation. e. Find a basis for the space of solutions to (1 − x2 ) d2 y dy + 9y = 0. −x dx2 dx 1.2.4. The diﬀerential equation − d2 + x2 z = λz dx2 (1.12) arises when treating the quantum mechanics of simple harmonic motion. a. Show that making the substitution z = e−x Hermite’s diﬀerential equation 2 /2 y transforms this equation into d2 y dy − 2x + (λ − 1)y = 0. 2 dx dx b. Show that if λ = 2n +1 where n is a nonnegative integer, (1.12) has a solution 2 of t...
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## This document was uploaded on 01/12/2014.

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