homework set 16 - BILKENT UNIVERSITY Department of...

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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 0 + 4 y = sin x, y ( - 2) = 2 , y 0 ( - 2) = 1 . b) y 00 + (cos x ) y 0 + 3(ln | x | ) y = 0 , y (2) = 3 , y 0 (2) = 1 . c) ( x - 2) y 00 + y 0 + ( x - 2)(tan x ) y = 0 , y (3) = 1 , y 0 (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 0 ) 2 = 0 , x > 0 . 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 ). 5) Verify that the functions y 1 and y 2 are solutions of the following given equations. Do they constitute a fundamental set of solutions. a) y 00 + 4 y = 0 , y 1 ( x ) = cos 2 x, y 2 ( x ) = sin 2 x.
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  • Spring '10
  • YosumHoca
  • #, Partial differential equation, general solution, two L, BILKENT UNIVERSITY Department of Mathematics MATH

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