# Math 230 - Chapter 8-9 Study Guide

- Find the general solution of the system
**X**'=**AX**. These are all 2×2 systems. Three parts.
- Use Variation of Parameters to solve. These are both 2×2 systems. The fundamental matrix
**Φ** is given. Two parts.
- Use the eigenvalues and eigenvectors from Maxima to write the general solution with real coefficients. These are both 3×3 systems. Two parts.
- Use RREF() on your calculator to find the general solution of the 3×3 system. The eigenvalues are given. Two parts.
- The formulas for the constants (k
_{1}, k_{2}, k_{3}, k_{4}) 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).
- Find the eigenvalues for a 3×3 system.
- Find the Matrix Exponential using Laplace Transforms and then find the general solution to the 2×2 system
**X**'=**AX**.
**Take Home**: 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.
- #8 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, although the Excel and Winplot files should be emailed to the instructor ahead of time.
- You may use a table of the common Laplace transforms on the exam.

## Points per problem

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

Pts |
24 |
16 |
12 |
12 |
12 |
6 |
8 |
10 |
100 |