# MATH 528 Write the non-homogeneous

December 7, 2016

Question
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. x2 y + 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)

MATH 528 HW3

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