2. Determine whether the following pair of functions are linearly dependent, or linearly independent:

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1 Topics to be covered on the exam include: Recognizing, and verifying solutions to homogeneous second-order linear differential equations, and their corresponding Initial Value Problems Recognizing and finding particular solutions to homogeneous, and non-homogeneous differential equation with constant coefficients by using either (1) the method of undetermined coefficients, or (2) the method of variation of parameters. Recognizing and solving homogeneous differential equations with variable coefficients if it s a Euler-type equation Solving problems involving Mechanical Vibrations, including those that are damped, undamped, free, and forced, and being able to recognize whether the vibrations are overdamped, critically damped, underdamped, or whether resonance is occurring. Be able to find the amplitude, frequency, and the phase shift of the cosine function which models the unforced, transient solution. Be able to answer theoretical questions from section 4.1 Sample Exam 1. Verify that y 1 = e x, y 2 = e 2x are solutions to the differential equation y 3y + 2y = 0. Now find the particular solution y = c 1 y 1 + c 2 y 2 given that y(0) = 1, y (0) = 0 [Section 4.1] 2. Determine whether the following pair of functions are linearly dependent, or linearly independent: (a) f(x) = sin 2 x, g(x) = 1 cos (2x) (b) f(x) = 1 + x, g(x) = 1 + x 3. Suppose we know that y(x) = c 1 e 10x + c 2 xe 10x is the solution to some equation ay + by + cy = 0. Find a, b, c. 4. The roots of a quadratic auxiliary equation are m 1 = 4, and m 2 = 5. What is the corresponding homogeneous linear differential equation? 5. The roots of a cubic auxiliary equation with real coefficients are m 1 = 1 2 and m 2 = 3 + i. What is the corresponding homogeneous linear differential equation?

2 6. Suppose y 1 = e 4x cos x is a solution of y + 6y + y = 34y = 0 What is the general solution of the differential equation? 7. Find a linear homogeneous constant-coefficient equation with the general solution y(x) = Ae 2x + Bcos 2x + C sin 2x. 8. Find the general solutions of 2y + 3y = 0 9. We know that the complementary solution y c = c 1 e x cos(x) + c 2 e x sin (x), and the particular solution y p = x + 1 are those of the non-homogeneous differential equation y 2y + 2y = 2x. Given the initial conditions y(0) = 4, and y (0) = 8, find the full solution. 10. Use the Wronskian to show that f(x) = 1, g(x) = x, and h(x) = x 2 are linearly independent on the real line. 11. Find the general solutions of the following differential equations: (a) y (4) = 16y (b) y + 5y + 5y = Given that y = e 3x is one of the solutions to y (3) + 3y 54y = 0, find the general solution. (HINT: Write out the auxiliary solution, then use synthetic division to reduce it to a quadratic.) 13. Given that y 1 = x 4 is a solution of the differential equation x 2 y 7xy + 16y = 0, find the general solution. HINT: Use technique of section 4.2

3 14. Find a linear homogeneous constant-coefficient differential equation having the general solution y(x) = (A + Bx + Cx 2 )e 2x 15. Determine the period and frequency of the simple harmonic motion of a body of mass 0.75 kg on the end of a spring with spring constant 48 N/m. 16. Find a particular solution y p of the following differential equations: (a) y + 16y = e 3x (b) y (3) + 4y = 3x 1 (c) y (5) + 5y (4) y = 17 (d) y (3) y = e x Solve the initial value problem y + 4y = 2x; y(0) = 1, y (0) = Use the method of variation of parameters to find a particular solution of the equation y + 3y + 2y = 4e x 19. A mass m = 2, with a spring constant k = 40, is attached to a dashpot with damping constant c = 16. The spring is first displaced to a position x 0 = 5, and given an initial velocity v 0 = 4. Determine whether the motion is overdamped, critically damped, or underdamped. Then find the function which describes its vibrations. Find the circular frequency, and the period of the oscillation.

4 20. A mass weighing 2 lb stretches a spring 6 in. If the mass is pulled down an additional 3 in, and then released, and if there is no damping, determine the position x(t) of the mass at any time t. Find the frequency, period, and amplitude of the motion. 21. Suppose the following differential equation describes the oscillations of a spring with a mass attached, along with a dashpot, and an external force: 1 5 x (t) + 1.2x (t) + 2x(t) = 5 cos 4t (a) What is the mass of the object in slugs? (b) What is the spring constant? (c) What is the damping constant? (d) What is the amplitude of the external force? (e) What is the circular frequency of the external force? 22. Suppose the differential equation x + 4x = cos ωt describes a motion of the spring that is forced. (a) Is damping taking place? (b) Supposing that the mass is 1 slugs, what is the spring constant? (c) What is the natural frequency ω 0? (d) At what frequency ω will resonance occur? 23. A 10-pound weight attached to a spring stretches it 2 feet. The weight is attached to a dashpot damping device that offers a resistance numerically equal to β (β > 0) times the instantaneous velocity. Determine the values of the damping constant β so that the subsequent motion is (a) overdamped, (b) critically damped, and (c) underdamped. 24. Suppose the forcing function is f(t) = 2 cos(3t) 3 sin(3t). Find the amplitude C, and the phase α of the forcing function: f(t) = C cos(ω α) 25. Solve the following Euler-type differential equations: (a) 2x 2 y + 3xy 15y = 0, y(1) = 0, y (0) = 1 (b) x 2 y + 3xy + 4y = 0

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