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10-22-11-b

# 10-22-11-b - Math 5-A FALL 2011 FINAL PRACTICE PROBLEMS(1...

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Math 5-A FALL 2011 FINAL PRACTICE PROBLEMS (1) Find the general solution of ( a ) 2 y 00 - 5 y 0 - 3 y = 0 , ( b ) y 000 + 3 y 00 - 4 y = 0 , (2) Find the general solution of ( a ) y 00 + 4 y = x cos( x ) , ( b ) y 00 - 9 y 0 + 14 y = x e 2 x + x, ( c ) y 00 + y = sec( x ) . (3) Find the solution following initial value problems ( a ) y 00 + 4 y = - 2 y ( π/ 8) = 1 / 2 , y 0 ( π/ 8) = 2 , ( b ) y 00 + 4 y = F 0 sin( ax ) , y (0) = y 0 (0) = 0 , ( c ) y 00 - 9 y 0 + 14 y = x e 2 x + x, y (0) = 1 , y 0 (0) = 0 . (4) Consider equation d 2 x dt 2 + 4 x = - 4 sin(2 t ) . (a) Find the solution satisfying the initial conditions x (0) = 0 , x 0 (0) = 1. (b) Sketch the graph of the solution for 0 t 4 π . (c) If the function x ( t ) in part (a) is the position at time t of a mass attached to a spring, how do you describe, in your own words, the motion of the mass? (Hint : Your answer may involve words such as amplitude, equilibrium point and pseudo-period. For a similar problem see Solutions First Midterm) (5) (a) Given the function x ( t ) = Ae - t + Be 3 t + e t + 3 , find a second order linear constant coefficients differential equation for which this function is the general solution. (SOLUTION : x 00 - 2 x 0 - 3 x = - 4 e t - 9) (b) If cos(2 t ) - sin(2 t ) = A cos(2( t - δ/ 2)). Find A and δ . (SOLUTION : A = 2 , δ = 3 π/ 2 + π/ 4) (6) Find the solution of the initial value problem x 00 ( t ) - x 0 ( t ) - x ( t ) = sin( t ) , x (0) = 1 , x 0 (0) = - 3 .

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10-22-11-b - Math 5-A FALL 2011 FINAL PRACTICE PROBLEMS(1...

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