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TEXAS A&M UNIVERSITY DEPARTMENT OF MATHEMATICS MATH 308-502 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. 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. Bonus question (+2pts): For which values of ω > 1 the solution is identically zero when t > π. Points: /25