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B ˙ ILKENT UNIVERSITY Department of Mathematics MATH 225, LINEAR ALGEBRA and DIFFERENTIAL EQUATIONS Homework set # 16 U.Mu˘gan July 16, 2008 FUNDAMENTAL SET OF SOLUTIONS 1) Find the Wronskian of the following given pair of functions: a) e 2 x , e- 3 x/ 2 . b) x, xe x . c) e x sin x, e x cos x . 2) Determine the largest interval in which the given I.V.P. is certain to have a unique twice differentiable solution. Do not find the solution. a) ( x- 1) y 00- 3 xy + 4 y = sin x, y (- 2) = 2 , y (- 2) = 1 . b) y 00 + (cos x ) y + 3(ln | x | ) y = 0 , y (2) = 3 , y (2) = 1 . c) ( x- 2) y 00 + y + ( x- 2)(tan x ) y = 0 , y (3) = 1 , y (3) = 2 . 3) Verify that y 1 ( x ) = 1 and y 2 ( x ) = x 1 / 2 are two L.I. solutions of yy 00 + ( y ) 2 = 0 , x > . Then show that c 1 + c 2 x 1 / 2 is not, in general a solution of the equation. Why not? 4) If Wronskian W of f and g is x 2 e x and f ( x ) = x , find g ( x ).... View Full Document