Mathematical Methods for the Physical Sciences I

MATH 528: Fall 2015 Due: 8 Sept 2015 (in class)

Homework 3

• Method of Undetermined Coefficients

– Find a general solution of the following:

1. y′′ + 5y′ + 4y = 2 cosh 2x (Hint: rewrite cosh 2x as a sum of exponentials). 2. y′′ + 5y′ + 4y = 2 cosh x. 3. y′′ + 4y′ + 4y = 2 cosh 2x.

• 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. x2y′′ + 3xy′ + y = 1 x + ln(x). (Hint: try the substitution x = et).

• Variation of Parameters

– Consider the ODE: (x2 − 1)y′′ − 2xy′ + 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)

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