MA2051 - Ordinary Differential Equations
Sample Exam Questions - C97

Originally Given 1996 D Term


Instructions: Do your work on the paper provided. Put your name and section number on the cover and on each page. Remember, your work and explanations are graded, not just the final answer.

1. (30 points) In solving the following problems, be sure to use work from previous parts if you can. As always, you must justify your answers.

Find two linearly independent solutions for tex2html_wrap_inline64 . Prove that your solutions really are linearly independent.
Find a particular solution for tex2html_wrap_inline66 .
Find the solution for tex2html_wrap_inline66 which satisfies tex2html_wrap_inline70 .

2. (30 points) Work with the following model for a damped, unforced spring:


When there is no damping (so p=0) you can assume that the general solution is of the form tex2html_wrap_inline76 where tex2html_wrap_inline78 is the natural frequency. Find the constants tex2html_wrap_inline78 , A and tex2html_wrap_inline84 for the motion if the mass is pulled down 6 cm and released from rest.

When the motion of the mass is opposed by a friction damping force (so p > 0), the general solution is not the same as in part (a). Find the smallest value for p that would eliminate all oscillations from the solution. (Note: you do not have to actually solve the differential equation to answer this question.)

Find the smallest damping coefficient p > 0 that will reduce oscillations with an initial amplitude of 6 cm to an amplitude of 0.01 cm in 60 seconds. Explain your method clearly.

3. (30 points) Consider the following initial value problem:


Give the characteristic equation and find its roots. Find two linearly independent solution pairs and use them to obtain the general solution for the system.

Find the solution satisfying the given initial conditions.

Sketch the solution curves for at least three initial points in the (y,z)-plane. Indicate clearly the direction of motion along the curve as t increases.

4. (10 points) Define the total energy of an unforced, damped (with damping coefficient p>0) spring-mass system by


Use the standard model for a damped spring to compute tex2html_wrap_inline100 and show that it is strictly negative (whenever the spring is moving) and so the energy is always decreasing.


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