Math questions due in 4 hours

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MATH 528: Fall 2015 Due: 8 Sept 2015 (in class) Homework 3 Forced Oscillations – Suppose that a car oscillates vertically as if it were a mass of m = 800 kg on a single spring with spring constant k = 7 104 N/m which is attached to a single damper with c = 3000 N s/m. Suppose that this car is driven along a sinusoidal road surface with an amplitude of 5 cm and a wavelength of L = 10 m. 1. Write the non-homogeneous ODE that governs the vertical motion of the car. (Make sure to keep track of units). 2. Find the steady-state amplitude of vibration of the car as a function of the horizontal driving speed v (m/s). 3. At what driving speed vm will the greatest amplitude of vertical vibrations occur? 4. If the driver would prefer that the car oscillates with a lower amplitude than that of the underlying road surface (i.e. A < 5 cm), in what range of speeds must they drive? Euler-Cauchy Equation – Find a general solution to the following: 1. x2y00 + 3xy0 + y = x1 + ln(x). (Hint: try the substitution x = et). Variation of Parameters – Consider the ODE: (x2 1)y00 2xy0 + 2y = x2 1. 1. Show that y1 = x is a solution to the associated homogenous problem. 2. Find a second, linearly-independent, solution to the homogeneous problem, y2, using the method of reduction of order. 3.Find a particular solution using the method of variation of parameters. Read Section 4.0 (Review of Matrix Concepts) MATH 528 HW3 Page 1 of 1
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