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# hw9 - t = 0(i Find the characteristic exponents r 1 and r...

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Math 135A, Winter 2008 Homework #9 (due 3/10/2008) Routine problems: § 9.4: #1, #3, #5, #7, #9,#11, #12, #14, #17, #21, #23, #27, #31, #33, #37, #38, #39; § 9.5: #1, #3, #8, #15. To hand in: (1) Problems 24 and 26 on page 1039. (2) Problem 28 on page 1039. (3) In each of the following, find the general power series soltion of the equation about t = 0. This includes finding a formula for the k th term of the series. (a) y 00 - ty 0 - y = 0 (b) y 00 + x 2 y = 0 (c) (2 + t 2 ) y 00 + 3 xy 0 + y = 0 (4) In each of the following, identify the singular point of the equation and tell whether they are regular or irregular. (a) 2 t 3 (4 - t 2 ) y 00 + 2 y 0 + 3 ty = 0 (b) ( t 2 - 1) 2 y 00 + t (1 - t ) y 0 + (1 + t ) y = 0 (c) The equation t (1 - t ) y 00 + ( c - ( a + b + 1) t ) y 0 - ab y = 0 ( a, b, c constants) is called the hypergeometric equation . Show that its singular points 0 and 1 are both regular, and find the characteristic exponents at each point. (To handle the point 1, make the change of variables x = t - 1.)
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Unformatted text preview: t = 0. (i) Find the characteristic exponents r 1 and r 2 (taking r 1 to be the larger one). (ii) Find a solution of the form y = ∑ ∞ a k t k + r with r = r 1 . (iii) Find another such solution with r = r 2 if possible or show that no such solution exists. (a) ty 00 + y = 0 (b) t 2 y 00 + ty-( t + 2) y = 0 (c) ( t-t 2 ) y 00 + (3-6 t ) y-6 y = 0 (d) ty 00 + (1-t ) y + λy = 0, ( λ = constant). Notes: (1) The equation in part (c) is a special case of the hypergeometric equation. (2) The equation of part (d) is called the Laguerre equation . You should ﬁnd that when λ is a nonnegative integer, the soltuion you get is a polynomial of degree λ . These polynomials are called Laguerre polynomials . 1...
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