# Math 230 - Chapter 8-9 Study Guide

- Determine whether the vectors form a fundamental set on (-∞,+∞). Two parts.
- Find the general solution of the system
**X**'=**AX**. These are all 2×2 systems. Three parts.
- Find the Matrix Exponential using Laplace Transforms and then find the general solution to the 2×2 system
**X**'=**AX**.
- Use the eigenvalues and eigenvectors from Maxima to write the general solution with real coefficients. Two parts.
- Find the general solution of the 3×3 system. The eigenvalues are given. The Use REF or RREF on your calculator as needed. Two parts.
- Find the Matrix Exponential using the definition.
- The constants for the Euler, Improved Euler, and Runge-Kutta 4 methods are given. Use them to find the next y value. You will need to know the formula for the next y value (including the weighting).
- Use Variation of Parameters to solve. Two parts.
- Use any of the techniques covered (except Maxima) to solve the 3×3 system
**X**'=**AX**.
- Use the Euler, Improved Euler, and RK4 methods to approximate the given value. Graph the actual solution and the approximations on the same graph.

## Notes

- You need to bring scratch paper for extra work, there is not enough room on the exam given for most of the problems.
- There is no Maxima allowed on this exam.
- #10 is the take home question for chapter 9. It is designed to be used with technology (Maxima, Excel, and Winplot). It is due at the beginning of the exam.
- You may use a table of the common Maclaurin series and a table of common Laplace transforms.

## Points per problem

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

Pts |
10 |
18 |
6 |
10 |
12 |
5 |
9 |
10 |
10 |
10 |
100 |