# Math 230 - Chapters 8-9 Study Guide

## In-Class Exam (Chapter 8)

- Write the linear system of differential equations in matrix form. Look
at problems 8.1.1-6
- Verify that the matrix
**X** is a solution of the given system.
Look at problems 8.1.11-16.
- Find the general solution to the system. Show work!
Look at problems 8.2.1-12.
- Find the general solution to the system. Use Derive to find eigenvalues
and eigenvectors. Two parts.
Look at problems in section 8.2.
- Find eigenvectors and the general solution to a system with repeated eigenvalues.
The eigenvalues are given.
Look at problems 8.2.19-28.
- Use the method of undetermined coefficients to solve the given system.
Look at problems 8.3.1-8.
- Use variation of parameters to solve the given system.
Look at problems 8.3.11-30.
- Compute the matrix exponential for the matrix
**A** using
the definition of matrix exponential and by matching it with known Maclaurin
series. Look at problems 8.4.1-4 .
- Compute the matrix exponential for the matrix
**A** by using
the Laplace transform to find the matrix exponential. Look at the Laplace
transform example in section 8.4.
- Use the matrix exponential to find the general solution of the non-homogeneous
differential equation. The matrix exponential is given.
Look at problems 8.4.9-12.

## Take Home Exam (Chapter 9)

This take home exam is due the day of the in-class exam. The take home exam
is worth 16 points.

For each differential equation below, use Excel to create
a spreadsheet that will compare the numerical methods of Euler's
method, the Improved Euler's
method, and the fourth order Runge-Kutta method similar to table 9.6 in section
9.2 of the text (but without the actual value) to approximate the function
at the
indicated point.

- y' = xy + sqrt(y), y(0) = 1; y(0.5)
- Create a table comparing Euler's method, Improved
Euler's method, and the fourth-order Runge-Kutta method
for h
= 0.1.
- Create a table comparing Euler's method, Improved
Euler's method, and the fourth-order Runge-Kutta method
for
h = 0.05.

- y' = x + y
^{2}, y(0) = 0; y(0.5)
- Create a table comparing Euler's method, Improved
Euler's method, and the fourth-order Runge-Kutta method
for h = 0.1.
- Create a table comparing Euler's method, Improved
Euler's method, and the fourth-order Runge-Kutta method
for h = 0.05.

## Notes

- A table of key Laplace transforms is provided with the test.
- A table of common Maclaurin series is provided with the test.

## Points per problem

# |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
TH |
Total |

Pts |
3 |
5 |
10 |
16 |
8 |
10 |
10 |
6 |
6 |
10 |
16 |
100 |