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Math. 2403, Practice Final 1. Solve the initial value problem dx t2 = , dt x + t3 x x(0) = −2. 2. A large tank is initially filled with 500 L of water containing 50 kg of salt. A brine containing 2 kg of salt per gallon is pumped into the tank at a rate of 5 L/min. The well-mixed solution is pumped out at the same rate. After 10 min, a leak develops in the tank and an additional liter per minute of mixture flows out of the tank. What is the concentration, in kilograms per liter, of salf in the tank 20 min after the leak develops? 3. Solve the initial value problem y 00 − y = sin t − e2t , y(0) = 1, y 0 (0) = −1. 4. Find a general solution of 0 0 x (t) = 0 −2 5. Let A= 1 0 −5 3 0 −2 3 0 1 x(t). −4 . (a) Find eAt . (b) Find the solution of 0 x (t) = Ax(t), 2 x(0) = . 3 (c) Draw the phase portrait for x0 (t) = Ax(t). 6. A mass-spring system contains two masses whose displacement functions x(t), y(t) satisfy the differential equations x00 = −4x + y, y 00 = x − 4y. Find the two natural frequencies of the system, its two natural modes of oscillation, and write down a general solution of the system. 7. Find the critical points of x0 = 2x − x2 − xy y 0 = 3y − 2y 2 − xy and classify their type and stability. Draw the phase portrait for the system. Indicate the directions of the vector field at various points. 8. Find L{f (t)}, where f (t) = ( e−t 1 for 0 < t < 1, for 1 < t < 2, and f (t) has period 2. 9. Use the method of Laplace transforms to solve the initial value problem ( y 00 + 5y 0 − 6y = 21et y(0) = −1, y 0 (0) = 9.