homework set 17 - BILKENT UNIVERSITY Department of...

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B ˙ ILKENT UNIVERSITY Department of Mathematics MATH 225, LINEAR ALGEBRA and DIFFERENTIAL EQUATIONS, Homework set # 17 U.Mu˘gan July 16, 2008 REPEATED ROOTS; REDUCTION OF ORDER 1) Find the general solution of the following D.E’s a) y 00 - 2 y + y = 0. b) 25 y 00 - 20 y 0 + 4 y = 0. c) y 00 - 2 y 0 + 10 y = 0. 2) Solve the following I.V.P. and describe the behavior of the solution for increasing x . a) y 00 + 4 y 0 + 4 y = 0 , y ( - 1) = 2 , y 0 ( - 1) = 1 . b) 9 y 00 - 12 y 0 + 4 y = 0 , y (0) = 2 , y 0 (0) = - 1 . c) 9 y 00 + 6 y 0 + 82 y = 0 , y (0) = - 1 , y 0 (0) = 2 . 3) Use the reduction of order to find the second L.I. solution of the following D.E. a) x 2 y 00 + 2 xy 0 - 2 y = 0 , x > 0 , y 1 ( x ) = x . b) xy 00 - y 0 + 4 x 3 y = 0 , x > 0 , y 1 ( x ) = sin x 2 . c) ( x - 1) y 00 - xy 0 + y = 0 , x > 1 , y 1 ( x ) = e x . 4) If y 1 is a known solution of y 00 + p ( x ) y 0 + q ( x ) y = 0, show that a second solution y 2 satisfies ˆ y 2 y 1 ! 0 = W
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homework set 17 - BILKENT UNIVERSITY Department of...

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