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Download Name________________________ Student I.D.___________________ Math 2250−1 Quiz 7
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Transcript
Name________________________ Student I.D.___________________ Math 2250−1 Quiz 7 October 21, 2011 SOLUTIONS 1) Consider the differential equation for y x y 5 y 6 y=0 . 1a) Find the general solution to this differential equation. rx rx trying y x = e yields L y = e of the characteristic polynomial 2 r 5 r rx 6 so in order for e (6 points) to be a solution, r must be a root r2 5 r 6 = r 3 r 2 . Since the roots are r = 3, 2 the general solution is yH x = c1 e 3 x c2 e 2 x . 1b) What is the dimension of the solution space above? 3x Since there are two basis functions, y1 = e 2x , y2 = e (1 point) for the solution space, the dimension is two. 1c) Use your work in (1a) to solve the initial value problem y 5 y 6 y=0 y 0 = 1 y 0 =4 . (3 points) y x = c1 e 3x c2 e 3x y x = 3 c1 e 2x 2 c2 e 2x at x = 0 : 1 = c1 c2 4 = 3 c1 2 c2 . 1 4 c1 c2 = 1 1 3 2 1 = 1 4 1 1 c1 3 2 c2 = 1 1 2 1 1 3 1 4 So y x = 2 e 3x e 2x . = 2 1 .
