Quiz 02 Solution - x p t ( ) = C , must resemble form of...

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ME 360 Fall 11 9/13/11 Name Quiz 2 Consider the differential equation  x + 5 x + 6 x = 6 1 t ( ) , with a unit step input and initial conditions x 0 ( ) = 0 , x 0 ( ) = 0 . (Unit step is zero for t < 0 , and one for t 0 .) a. Find the solution to the differential equation, being sure to follow the procedure with homogeneous and particular solutions. b. Suppose the left-hand side of the equation were  x + 2  x +  x + 5 x + 6 x = 0 . How many exponential terms would be expected in the homogeneous solution? c. When should the initial conditions be plugged in to find the unknown coefficients of the homogeneous solution? Before or after the particular solution has been added? (choose one) (i) After (ii) After (iii) After (iv) After (v) After! Characteristic equation: s 2 + 5 s + 6 = 0 s 1,2 = 2, 3 (e.g. from quadratic formula or completing the square) Homogeneous solution x h t ( ) = A 1 e 2 t + A 2 e 3 t , A 1 and A 2 unknown for now Particular solution
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Unformatted text preview: x p t ( ) = C , must resemble form of right-hand side, which is 0 plug into equation to find C : C + 5 C + 6 C = 6 , C = 1 , x p t ( ) = 1 Genera solution is sum of two. x t ( ) = x h t ( ) + x p t ( ) = A 1 e 2 t + A 2 e 3 t + 1 Now plug in initial conditions x ( ) = A 1 e 2 + A 2 e 3 + 1 = A 1 + A 2 + 1 = x ( ) = 2 A 1 e 2 t t = 3 A 2 e 3 t t = = 2 A 1 + 3 A 2 = A 1 = 3 2 A 2 , 3 2 A 2 + A 2 + 1 = 1 2 A 2 = 1 , A 2 = 2 , A 1 = 3 a. x t ( ) = 3 e 2 t + 2 e 3 t + 1 for t b. For an n th order differential equation, the characteristic equation will have n roots, and so there will be n (complex) exponentials in the homogeneous solution. c. (i-v) After. Always remember to plug in initial conditions at the very end....
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This note was uploaded on 11/22/2011 for the course ME ME360 taught by Professor Kuo during the Fall '11 term at University of Michigan.

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