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midrev2 - (7 y 00 16 y = tan 4 x Hint variation of...

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MAP 2302 Fall 2010 Review for Midterm 2 Format: 5 problems, 20 points each. Time allowed: 50 minutes. Second Order Linear Equations and Laplace Transform: general solutions, equations with constant coefficients, Cauchy-Euler equations, method of undetermined coefficients, variation of parameters, initial-value problems, Laplace transforms of table functions, properties, inverse transforms using partial fraction decomposition. In problems (1–3), solve the initial-value problem: (1) t 2 x 00 - 6 x = 0 , t > 0 , x (1) = 1 , x 0 (1) = 2 . ( Hint: Cauchy-Euler ) (2) y 00 + 3 y 0 + 2 y = x 2 , y (0) = 0 , y 0 (0) = 1 . ( Hint: undetermined coefficients ) (3) y 00 + y = g ( x ) , y (0) = 1 , y 0 (0) = 0 provided that y p ( x ) = ( x + 1) 2 is a particular solution. In problems (4–6), find the general solution: (4) y 00 + 16 y = xe x . ( Hint: undetermined coefficients ) (5) xy 00 + 3 y 0 + 2 x y = 0 , x > 0 . ( Hint: Cauchy-Euler ) (6) y 00 - y = sin x - 2 cos x . ( Hint: undetermined coefficients ) In problems (7–8), find a particular solution:
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Unformatted text preview: (7) y 00 + 16 y = tan 4 x. ( Hint: variation of parameters ) (8) xy 00 + y-4 x y = x 3 , x > 0. ( Hint: homogeneous part is Cauchy-Euler ) (9) Determine the form of a particular solution (do not evaluate the coefficients) for the equation y 00 + 6 y + 10 y = xe-3 x cos x + cos 2 x. In problems (10–12), determine the Laplace transform of the given function: (10) f ( t ) = 7 e 2 t cos(3 t )-2 e 7 t sin(5 t ) . (11) f ( t ) = ( t + 3) 2-( e t + 3) 2 . (12) f ( t ) = ± e-t , ≤ t ≤ 2 , 2 , t > 2 . In problems (13–14), determine the inverse Laplace transform of the given function: (13) F ( s ) = 2 s-1 s 2-4 s + 6 . (14) F ( s ) = 2 s 2 + 3 s-1 ( s + 1) 2 ( s + 2) . We will discuss these problems in class on 10/20/10. The key will be posted separately....
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