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TEXAS A&M UNIVERSITY DEPARTMENT OF MATHEMATICS MATH 308-200 Exam 2 version A, 31 Oct 2014 On my honor, as an Aggie, I have neither given nor received unauthorized aid on this work. Name (print): In all questions no analytical work =⇒ no points! 1. Solve the initial value problem y 00 − y 0 − 6y = e3t , using the method of your choice. y(0) = 1, y 0 (0) = 0, 2. Find the general solution to y 00 + 2y 0 + y = t−2 e−t . 3. A series circuit has a capacitor of C = 0.25 × 10−6 F, a resistor of R = 5 × 103 Ω, and an inductor of L = 1H. The initial charge on the capacitor is zero. A 12-volt battery is connected to the circuit and the circuit is closed at t = 0. What’s the charge on the capacitor after a long time? Give a quick physics explanation and a full proof using the solution of the differential equation LQ00 + RQ0 + 1 Q = E(t). C 4. Prove that the Laplace transform of the function sin(t), 0 < t < π, g(t) = 0, t≥π is equal to 1 1 + e−πs . +1 You may do it using the definition of Laplace transform or by expressing g(t) in terms of the Heaviside function u(t) and using the formulas. Bonus (+2pnts): Do it using both methods. Do not attempt to fudge your answer! G(s) = s2 5. Solve the initial value problem y 00 + ω 2 y = g(t), ω > 1, y(0) = 0, y 0 (0) = 0, where g(t) is given in question 4. For which values of ω > 1 is the solution identically zero when t > π. Points: /25