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TEXAS A&M UNIVERSITY DEPARTMENT OF MATHEMATICS MATH 308-200 Exam 1 version A, 29 Sep 2011 On my honor, as an Aggie, I have neither given nor received unauthorized aid on this work. Name (print): No detailed analytical work — no points. 1. Find the general solution of 2 xy 0 − (2x2 + 1)y = −ex . 2. Find the general solution to 1 3 y 0 = 0; 3x sin(2y) + 2x cos(2y) + y 2 3. For the equation y 0 = y(1 − y), 1. Find stationary solutions (critical points) and draw the phase diagram. 2. Using the phase diagram, predict the large t limit of the solution y(t) that satisfies the initial condition y(0) = 1/2. 3. Find explicit solution to the IVP and compare with your prediction. 4. The small oscillations of a pendulum are described by the equation θ00 + bθ0 + 16θ = 0, where θ(t) is the angle and b is related to friction. 1. Find the general solution for b = 0, 4, 8, 10. 2. For which of the above values the solution tends to 0? Explain the physical meaning of your answer. 3. For which of the above values the solution y(t) will keep changing sign (i.e. y(t) = 0 has infinitely many solutions). Would your answer depend on the initial conditions? 5. (Bonus question: 2pnts) Solve ty 00 − y 0 − 4t3 y = 0, by looking for solution of the form y(t) = f (t2 ) and obtaining an equation in terms of f (u) and u (where u = t2 ). Points: /20