##
MA 231:
Honors Differential Equations
Final Exam

Do all problems on the blank sheets in order. Please
write the problem number in the upper right hand corner of each sheet.
Be sure to show all work.
1. Quickies. Answers only---no partial credit.

- Give an example of a point that has a dense orbit in the
interval
**0 <= x <= 1** under
iteration of **10x mod 1**.
- Sketch the slope field and solution graphs for

**y' = y**^{3} if y >= 0, or
=y(y+1) if y < 0
- Sketch the phase line for
** y' = cos y**.
- Give the second order differential equation for a damped
harmonic oscillator with mass 1, spring constant 2, damping constant
3, and forcing function 4.
- Find one solution of the nonlinear second order equation
**
y'' + tan**^{3} y = 0.
- Sketch the phase plane for the linear system
**
x' = pi**^{{666}1/2}x

y' = -e^{1.4 pi}y
- Give an example of a 2 by 2 matrix whose eigenvalues are
** + or - {7}**^{1/2}i.
- Draw a graph of a function that has a neutral fixed point at
0, a repelling fixed point at 1, and an attracting fixed point at
-1.
- State the Existence and Uniqueness theorem for first order
differential equations.
- Give an example of a first order differential equation that
has no equilibrium points whatsoever.

2. Prove that the Laplace transform of the Heaviside
function **u**_{a}(t) is **Y(s) = e**^{{-as}}/s.
Show ** all** steps.

3. Find the general solution of the linear system
**
Y' = AY where A =**

(-1 1)

(0 -2)

Then sketch the phase plane for this system.

4. In an essay of no more than two pages, discuss the
meaning of the trace-determinant plane. Be sure to include a picture
and a discussion of why this picture is useful.

5. Use graphical iteration to describe the fate of a
representative collection of orbits
of **F(x) = 2x(1-x)** for seeds **0 < x < 1**.

6. Solve the differential equation

**
y'' + y = 7 if t < 3, or =
0 if t >= 3.
**
with **y(0)=0** and **y'(0) = 0** using Laplace transforms.
7. Consider the system of equations

**
x' = x - y**^{2}

y' = y - x^{2}

First find all equilibirum points. Determine the type of these
equilibrium points.
Sketch the nullclines and indicate
the direction of the vector field on and in between these curves.
Finally, sketch the solution curves of this system.