# Math 230 - Chapter 8-9 Study Guide

1. Determine whether the vectors form a fundamental set on (-∞,+∞). Two parts.
2. Find the general solution of the system X'=AX. These are all 2×2 systems. Three parts.
3. Find the Matrix Exponential using Laplace Transforms and then find the general solution to the 2×2 system X'=AX.
4. Use the eigenvalues and eigenvectors from Maxima to write the general solution with real coefficients. Two parts.
5. 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.
6. Find the Matrix Exponential using the definition.
7. 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).
8. Use Variation of Parameters to solve. Two parts.
9. Use any of the techniques covered (except Maxima) to solve the 3×3 system X'=AX.
10. 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

 # Pts 1 2 3 4 5 6 7 8 9 10 Total 10 18 6 10 12 5 9 10 10 10 100