Mathematics 304, section 2 – Ordinary Differential Equations Midterm Exam 2 November 19, 2004

Directions

Read each question carefully before starting to work. Do all work in the blue exam booklet. There are 100 regular points and 10 Extra Credit points (see back).

I. All parts of this question refer to the family of 1st order systems

(1)  X′ = ( \begin{matrix}−1 & a \\ −1 & 0 \end{matrix} ) X

depending on the parameter a. A) (5) What type of equilibrium point does the system in this family with a = 10 have

at the origin? (You do not need to find the general solution for this one!) B) (20) Find the general solution of the system in the family with a = −2. What type

of equilibrium point does this system have at the origin? C) (5) Sketch the motion in the trace-determinant plane corresponding to the family (1)

as a ranges over all real values. Besides the cases from parts A and B, what other types of phase portraits are possible in this family?

II. A) (10) Show that if x_{p,1} is a particular solution of mx′′ + bx′ + kx = g_{1}(t) and x_{p,2} is

a particular solution of mx′′ + bx′ + kx = g_{2}(t), then the sum x_{p,1}(t) + x_{p,2}(t) solves mx′′ + bx′ + kx = g_{1}(t) + g_{2}(t).

B) (20) Consider the following initial value problem: x′′ + 4x′ + 3x = e^{7t} + 4e^{−t}, with x(0) = 0, x′(0) = 1. Find the solution using part A and the method of undetermined coefficients.

III. (15) For what value of ω will resonance occur in solutions of 4x′′ + 400x = cos(ωt)? Explain.

IV. (10) A 4 × 4 matrix A has eigenvalues λ = 1, 2, each with multiplicity at least one, and no other eigenvalues. A satisfies dim ker(A − I) = 1 and dim ker(A − 2I) = 2. What can you say about the canonical form of A?

V. (15) Find all equilibrium points of the system:

(2)  \{ \begin{matrix}\frac{dx}{dt} = −y(1 − y) \\ \frac{dy}{dt} = −2x(1 + x) \end{matrix}

(over for Extra Credit)

Extra Credit (10) The following is a plot of a direction field and a solution for some autonomous first order system.

Can this be the system from (2) on previous page? Explain how you can tell.