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solfinal - MAT 127 PRACTICE FINAL(1 Consider the initial...

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MAT 127 PRACTICE FINAL (1) Consider the initial value problem y 00 - y 0 + 3 y = 0 y (0) = 1 , y 0 (0) = - 1 . Assuming the solution to this initial value problem has is the power series y = X n =0 c n x n , find all the coeffiecients c n for n 6. Solution: We have that y 00 = X n =0 ( n + 2)( n + 1) c n +2 x n , - y 0 = X n =0 - ( n + 1) c n +1 x n , 3 y = X n =0 3 c n x n . If we add these series together (adding term by term) we must get the power series n =0 0 x n (this is what the differential equation states). Thus, for all n 0 we have that (1) ( n + 2)( n + 1) c n +2 - ( n + 1) c n +1 + 3 c n = 0 . On the other hand, we deduce from y (0) = 1 and from y 0 (0) = - 1 that (2) c 0 = 1 and (3) c 1 = - 1 . To compute c 2 we use properties (1)-(3) (with n = 0 in (1)) to get 2 c 2 - ( - 1) + 3 = 0 from which we can solve for (4) c 2 = - 2 . To compute c 3 we use properties (1),(3),(4) (with n = 1 in (1)) to get 6 c 3 - 2( - 2) + 3( - 1) = 0 1
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2 MAT 127 PRACTICE FINAL from which we can solve for (5) c 3 = - 1 / 6 . To compute c 4 we use properties (1)(4)(5) (with n = 2 in (1)) .....
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