Pde

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Unformatted text preview: 1 1 x− x3 + x4 − · · · 3·2 (5 · 3)(3!) (7 · 5 · 3)4! √ 1 1 1 +c2 x 1 − x + x2 − x3 + · · · . 3 2·5·3 3! · (7 · 5 · 3) We obtained two linearly independent generalized power series solutions in this case, but this does not always happen. If the roots of the indicial equation differ by an integer, we may obtain only one generalized power series solution. In that case, a second independent solution can then be found by variation of parameters, just as we saw in the case of the Cauchy-Euler equidimensional equation. Exercises: 1.3.1. For each of the following differential equations, determine whether x = 0 is ordinary or singular. If it is singular, determine whether it is regular or not. a. y + xy + (1 − x2 )y = 0. b. y + (1/x)y + (1 − (1/x2 ))y = 0. c. x2 y + 2xy + (cos x)y = 0. d. x3 y + 2xy + (cos x)y = 0. 1.3.2. Find the general solution to each of the following Cauchy-Euler equations: a. x2 d2 y/dx2 − 2xdy/dx + 2y = 0. b. x2 d2 y/dx2 − xdy/dx + y = 0. c. x2 d2 y/dx2 − xdy/dx + 10y = 0. (Hint: Use the formula xa+bi = xa xbi = xa (elog x )bi = xa eib log x = xa [cos(b log x) + i sin(b log x)] to simplify the answer.)...
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This document was uploaded on 01/12/2014.

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