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MATH 2351 Ordinary Differential Equations Spring 2013 Quiz 2 1. 8 points Consider the following predator-prey system: dx xy = −x + , dt 2 dy = 2y(7 − y) − 5xy. dt (a) Identify which variable represents the predator population, and which one represents the prey population. Which ones are larger, the predators or the preys? (b) Find all the equlibrium solutions of this system. 2. 8 points Conside the following (partially decoupled) system: (a) Find the general solution of this system. (b) Find the solution satisfying x(0) = 1, y(0) = 0. dx = −4x, dt dy = 2x + 3y. dt MATH 2351 Quiz 2 Spring 2013 3. 4 points (a) Convert the differential equation y 00 (t) = 3y 0 (t) + 8y(t) − 5y 2 (t) to a system of first order differential equations. You DO NOT have to solve the system. (b) Write the system dx1 dx2 = 2x1 − 7x2 , = 9x1 + x2 in matrix form. dt dt 3 2 . 6 −1 4. 10 points Consider the linear system dX/dt = AX, where A = −1 1 −3t 5t (a) Verify that X1 (t) = e and X2 (t) = e are solutions to this system. 3 1 (b) Verify that X1 (0) and X2 (0) are linearly independent vectors in the plane. 0 (c) Find the solution X(t) with the initial value X(0) = . 4