# Math 230 - Chapter 8-9 Study Guide

- Find the general solution of the system \( \mathbf{X}' = \mathbf{A} \mathbf{X} \)
**.** These are all 2×2 systems. Three parts.
- Find a 2×2 matrix \(\mathbf{A}\) so that \( \mathbf{X}' = \mathbf{A} \mathbf{X} \) has the indicated eigenvalues.
- Find the Matrix Exponential using Laplace Transforms. \( (s\mathbf{I}-\mathbf{A} )^{-1} \) is given but you will need to use partial fractions.
- Use Variation of Parameters to solve the 2×2 initial value problem. The fundamental matrix \( \mathbf{\Phi} \) is given.
- Use the eigenvalues and eigenvectors from Maxima to write the general solution to the 3×3 system using real coefficients only.
- Use RREF() on your calculator to find the general solution of the 3×3 system with repeated eigenvalues. The eigenvalues are given.
- Use the Euler, Improved Euler, and RK4 methods to approximate the solution to an initial value problem. The number of steps required is inversely proportional to the complexity of each step (you'll have more steps with Euler than with RK4). In all cases, the number of required steps is small; enough to show that you know what you're doing but not take a lot of time. Then use Laplace transforms to find the actual solution and find the relative error (as a percent) in the approximations using numeric methods.
**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

- There should be room on the exam to work the questions, but you may want to bring scratch paper.
- #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 should know the big-4 of the common Laplace transforms.

## Points per problem

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

Pts |
30 |
7 |
7 |
10 |
10 |
10 |
16 |
10 |
100 |