MATH 134, FALL 2007PRACTICE QUESTIONS FOR MIDTERM 2

1. Compute the flow φt(X) of the equation X ′ = AX, where

A =

[0 1

−4 0

].

2. Let

A =

[−1 −1

1 −1

], B =

[−1

2−5

212−3

2

].

Are the flows of X ′ = AX and Y ′ = BY topologically conjugate?

3. Consider the undamped forced harmonic oscillator

x′′ + x = 1.

(a) Convert the given second order equation into a non-homogeneous planar linear

system.

(b) Compute the solution to the system corresponding to the initial condition x(0) =

0, x′(0) = 0.

4. If φt(X) is the flow of X ′ = F (X), show that φct(X) is the flow of X ′ = cF (X), where

c is a nonzero constant.

5. Suppose that the flows of X ′ = AX and Y ′ = BY are topologically conjugate. If A is

a hyperbolic matrix such that all solutions of X ′ = AX tend to the origin, as t→∞,

and

B =

[a− 3 5

−2 a + 3

],

show that a < 0.

6. Consider the system

x′ = x + y2

y′ = 2y.

(a) Find the unique equilibrium.

(b) Linearize the system at the equilibrium and the describe the phase portrait of

the linearization.

(c) Describe the phase portrait of the nonlinear system.

(d) Does the linearized system accurately describe the local behavior near the equi-

librium?1

2 MATH 134, FALL 2007 PRACTICE QUESTIONS FOR MIDTERM 2

7. Suppose that h : R→ R is a smooth (i.e., differentiable as many times as you want)

conjugacy between the ODEs

x′ = αx and y = βy.

Show that α = β.