# Exam #3 Review

## Scheduling Information:

Time: Monday, April 17, 7:30-8:20AM.
Location: DH 2315.
Review Session: TBA. (Tentatively Saturday Afternoon)

## Topics:

Section 7.5.
Section 7.6.
Systems of Equations.
Parametric Curves and the Phase Plane.
Laplace Transform Applied to Systems.
Nullclines.
Graphs of solutions and phase portraits.
Classification of equilibrium points.

## Review Quesionts:

1. What is the Dirac delta function?
2. Why is the Dirac delta fiction not really a function?
3. What is \int_{-\infty}^\infty f(t)\delta(t-c)dt?
4. What is a system of differential equations?
5. When is a system of differential equations "linear"?
6. When is a system of differential equations "homogeneous"?
7. When is a system of differential equations "constant coefficient"?
8. How do you use the Laplace Transform to solve a linear, homogeneous, constant coefficient, (2-dimensional) system of differential equations?
9. What is the phase plane for a 2-dimensional system of differential equations?
10. How can you use analytic and qualitative methods to sketch trajectories for a 2-dimensional, first order, constant coefficient system of equations when the roots of the characteristic polynomial are complex? Distinct real? Repeated real?
11. What is a "phase portrait" for a 2-dimensional, first order, linear system of equations?

## Exercises:

Section 7.5 #1, 3, 5, 11, 15.
Section 7.6 #1, 3, 7.
Section 8.2 #1, 3, 19, 33, 35. [Where the instructions say "find the general solution" you should instead (a) choose an initial condition and use the Laplace transform to find the corresponding solution, (b) plot the nullclines and indicate where solutions are travelling left or right, up or down, or up-right, up-left, down-left, down-right, (c) on a separate set of axis sketch a phase portraait for the system (you can include the nullclines in this diagram, too), (d) classify the equilibrioum point (type and stability).]

## Old Exam Problems:

Here is a collection of exam problems I've given to differential equations students in previous semesters.

And here is the table of Laplace transforms you will be given on the exam. The numbers on the table correspond to the entries in Table 6.2.1.