MATH 331 MIDTERM REVIEW - PRACTICE PROBLEMS Chapter 1 (Sections 1.1-1.5, 1.7): 1. Suppose the population of a small island was 1000 in the year 2000, and it became doubled in the year 2010. Use the exponential growth model to predict the population of the island for the year 2020. 2. Consider the ODE y 0 = (1−y)2 (y +2)(y −3). (a) Find all the equilibrium solutions. (b) If y(x) is a solution with y(0) = 0, is y(x) an increasing function or a decreasing function? (c) For what initial values a solution is always increasing? 3. Solve separable ODEs. (1) y 0 = (x + 1)y 2 , (2) y 0 = (3) y 0 = y 2 − 1, (4) y 0 = x , y(0) = −1. x2 y+y 2 (y +1)x , y(0) = 1. y 4. Suppose y(x) = sin x, 0 ≤ x ≤ π/2, is a solution of ODE y 0 = f (y). Find a formula for the function f (y)? 1 1 and y2 (x) = 1−x are solutions. (2) 5. Consider ODE y 0 = y 2 . (1) Show that y1 (x) = 10−x 1 Suppose y(x) is a solution with initial value obeying 10 < y(0) < 1, using the Uniqueness Theorem to show that 1 1 < y(x) < . 10 − x 1−x (3) Redo part (2) by solving the equation, instead of using the Uniqueness Theorem. 6. (i) Sketch the direction fields for the ODE y 0 = y + x − 1. (ii) Based on the direction fields, explain why y = −x is a solution. (iii) Based on the direction fields, is the solution with initial value y(0) = 1 an increasing function or a decreasing function? 7. Solve first order linear ODEs. y (1) y 0 = x+1 + x2 + x + 1, y(0) = 1, (2) y 0 + 2 (3) y 0 = 2y + x3 e x , (4) y 0 + 2xy = e−x . x y x = e−x /x, 8. Exact ODEs and Integration Facters. (1) Solve 2xydx + x2 dy = 0, (2) Solve −ydx + xdy = 0. 9. A tank contains 80 lb of salt dissolved in 500 gal of water. The inflow per minute is 20 lb of salt dissolved in 20 gal of water. The outflow is 20 gal/min of uniform mixture. Find the time when the salt content y(t) in the tank reaches 95% of its limiting value as t → ∞. Chapter 2 (Sections 2.1, 2.2, 2.4, 2.7, 2.8): 10. For each of the following ODEs: (i) y 00 + 4y 0 + 3y = 0, (ii) y 00 + 4y 0 + 4y = 0, (iii) y + 4y 0 + 5y = 0, (1) find the general solution, (2) solve initial values: y(0) = 1, y 0 (0) = 0. 00 1 2 MATH 331 MIDTERM REVIEW - PRACTICE PROBLEMS 11. Consider free oscillations of a mass-spring system: my 00 + cy 0 + ky = 0, where (i) m = 1, c = 0, k = 4, (ii) m = 2, c = 2, k = 5. Classify the oscillator as underdamped, overdamped, critically damped, or undamped. 12. For the following ODEs: (i) y 00 + 4y 0 + 4y = ex , (ii) y 00 + 4y 0 + 4y = e−2x , (iii) y + 4y 0 + 4y = x2 , (1) find the general solution, (2) solve initial values: y(0) = 1, y 0 (0) = 0. 00 13. Consider an undamped mass-spring system, where a ball of mass 1 kg is attached to a spring which is stretched down by 2.45 cm in the equilibrium state. (1) Find the natural frequency of the system. (2) Find the motion of the ball if the initial position is at the equilibrium point and the initial velocity is 10 cm/sec. 14. Consider a critically damped mass-spring system where a ball of mass m = 1 is attached to a spring, and where the damping constant c = 2. Suppose the initial position y(0) = 2 and initial velocity y 0 (0) = −4. Will the ball ever cross the equilibrium point? Explain why. 15. Consider the damped forced oscillation y 00 + 4y 0 + 13y = cos 3t. (1) Find the general solution. (2) Write the ”steady-state” yp (t) as A cos(3t + φ), and find the amplitude A and phase φ. (3) Solve initial value: y(0) = 0, y 0 (0) = 0.

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