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MATH 308 EXAM 2 REVIEW (1) Solve the initial value problem. Describe the behavior of the solution for increasing t: growing oscillation, decaying oscillation, steady oscillation, or approaching ∞, −∞, 0? (a) y 00 + 4y = 0, y(0) = 1, y 0 (0) = 3 (b) 2y 00 − y 0 + y = 0, y(0) = 2, y 0 (0) = 0 1 2 (c) 9y 00 + 12y 0 + 4y = 0, y(0) = −1, y 0 (0) = 1 3 (2) Given that y1 (t) = et is a solution of (t − 1)y 00 − ty 0 + y = 0, t > 1, find a second solution by using the method of reduction of order. 4 (3) Find the general solution of the given differential equation. (a) y 00 + 4y = t2 + 3et (b) y 00 + 2y 0 = 4 sin 2t 5 (c) 4y 00 + y = 2 sec t , 2 −π < t < π 6 (4) A mass of 1kg is attached to a spring which has a spring constant of 4 N/m. The mass is pulled down 2 meters from its equilibrium position and set in motion with an initial velocity 1 m/s. Assume there is no external force. (a) If there is no damping, determine the position u(t) of the mass at any time t. Find the frequency, period, amplitude, and phase of the vibration. (b) If the vibration takes place in a viscous medium with a damping constant of 2 N·s/m, determine the position u(t) of the mass at any time t. Find the quasi-frequency, quasi-period, and phase of the vibration. (c) Determine the value of the damping constant for which the system is critically damped. (d) Determine the values of the damping constant for which the system is overly damped. 7 (5) A mass of 2kg is attached to a spring with a spring constant of 6 N/m. (a) Suppose there is no damping, and the mass is acted on by an external force of 5 cos ωt newtons. Determine the value of ω for which the resonance occurs. (b) Suppose the vibration takes place in a viscous medium with a damping constant of 4 N·s/m, and the mass is acted on by an external force of 2 sin 2t newtons. If the mass is set in motion from its equilibrium position with an initial velocity 3 m/s. Determine the position u(t) of the mass at any time t. Identify the transient and steady state parts of u(t). 8 (6) Use the definition to compute the Laplace transform of the function f (t) = eat cos bt (7) Express f (t) in terms of the unit step function uc (t). 2 if 0 ≤ t < 3 t , f (t) = 9, if 3 ≤ t < 5 3t − 1, if t ≥ 5 9 (8) Find the inverse Laplace of the given function. 2s − 1 (a) F (s) = 2 s − 4s − 5 (b) F (s) = 2s2 − s + 4 s3 + 4s 10 (c) F (s) = s2 (s + 2)3 (d) F (s) = (s + 8)e−5s s2 + 4s + 13 11 (e) F (s) = (s + 1)e−2s s2 + 2s − 3 (f) F (s) = 3e−s 2e−3s 2e−5s + − s s4 (s − 2)3 12 (9) Solve the following initial value problem. (a) y (4) − 16y = 0, y(0) = 0, y 0 (0) = 0, y 00 (0) = 1, y 000 (0) = 2 13 (b) y (4) − y = f (t), y(0) = 0, y 0 (0) = 0, y 00 (0) = 0, y 000 (0) = 0, 0, if 0 ≤ t < 1 f (t) = 1, if 1 ≤ t < 2 0, if t ≥ 2 where 14 (c) y 00 − y = f (t), y(0) = 0, y 0 (0) = 0, where ( sin t, if 0 ≤ t < π f (t) = 0, if t ≥ π 15 (d) y 00 − y = −20δ(t − 3), y(0) = 1, y 0 (0) = 0 π (e) y 00 + 4y 0 + 5y = δ t − sin t, 6 y(0) = 0, y 0 (0) = 0