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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- Fall '08
- Math, Elementary algebra, singular point, hypergeometric equation, characteristic exponents