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math16B_review_exer2-1

# math16B_review_exer2-1 - Math 16B Sarason REVIEW EXERCISES...

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Math 16B Spring 2011 Sarason REVIEW EXERCISES 2 1. Perform the integrations. (a) Z tan x cos 3 x dx (b) Z tan x cos 3 x dx (c) Z 1 0 x ( x - 1) 4 dx (d) Z 1 0 x 2 ( x - 1) 5 dx (e) Z 4 0 e - x dx (f) Z 1 x π - 5 dx (g) Z 1 x - 1 . 0001 dx (h) Z 1 x - e dx 2. Find the general solution of the differential equation y 0 = - 3 y 2 t and the solutions satisfying the initial conditions y (1) = 1 and y (1) = 0. 3. Find the general solution of the differential equation 2 yy 0 = - ( y 2 - 1) 2 and the solutions satisfying the initial conditions y (1) = - 2 and y (1) = - 1. 4. Find the general solution of the differential equation y 0 = - ( t - 1) 2 ( y + 1) 4 and the solutions satisfying the initial conditions y (0) = 1, y (0) = 0, y (0) = - 1. 5. Find the general solution of the differential equation y 0 = y 5 t . 6. Find the solutions that are defined for all t of the differential equation y 0 = e - y cos t . 7. Find the general solution of the differential equation y 0 = t (1 + y ) in two ways: by separation of variables, and by the method for solving linear first-order ODEs.

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