practicefinal.pdf - Math 286 Spring 2016 Practice Final...

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Math 286 Spring 2016 Practice Final Exam (1) Solve the initial value problem, where primes denote derivatives with respect to x , ( y 0 = y sin x, y (0) = 1 . (2) Find the general solution of the equation y 0 + 3 x 2 y = e - x 3 sin x. (3) Suppose that the characteristic equation of a differential equation is ( r 2 + 1)( r - 2) 2 = 0 . (a) Find the differential equation associated to this characteristic equation. (b) Find the general solution of the differential equation obtained from (a). (4) Consider an undamped system consisting of a mass of 4 kg on a spring with constant k = 4 N/m, and external force F ( t ) = 2 sin ωt . (a) Let ω = 2. Set up the nonhomogeneous equation which describes this system and find a particular solution. (b) Find the external frequency ω which causes resonance. (5) Let A = 1 1 1 1 1 1 1 1 1 . (a) Find all eigenvalues of A and their defects. (b) Find the general solution of the system ~x 0 ( t ) = A~x ( t ) . (6) Find a particular solution of the differential system ( x 0 = 4 x + y - 1 , y 0 = x + 4 y - e t . 1
2 (7) (a) Suppose that f is a function of period 2 with f ( t ) = t for 0 < t < 2 . Show that f ( t ) = 1 - 2 π X n =1 sin( nπt ) n (b) Substitute an appropriate value of
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