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Unformatted text preview: MTHSC 208 (Differential Equations) Dr. Matthew Macauley HW 15 Due Monday March 9th, 2009 (1) Find the general solution of x 2 y 00- xy- 3 y = 0. (2) Find the general solution of x 2 y 00- xy + 5 y = 0. (3) Find the general solution of x 2 y 00- 3 xy + 4 y = 0. (4) Consider the following ODE: y 00- 2 xy + 10 y = 0. (a) Assume the solution has the form y ( x ) = X n =0 a n x n . Plug y ( x ) back into the ODE and find the recurrence relation for a n +2 in terms of a n . Write down the general solution of the ODE. (b) Explicitly write out the coefficients a n for n 9, in terms of a and a 1 . See the pattern? Write down the formula for a n in terms of a and a 1 . (c) Find a basis for the solution space of the ODE (functions y ( x ) and y 1 ( x ) such that the general solution is y ( x ) = C y ( x ) + C 1 y 1 ( x )). (d) Find a non-zero polynomial solution for this ODE. [ Hint : Make a good choice for a and a 1 .] (e) Are there any other polynomial solutions, excluding scalar multiples of the one you...
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