# Math 230 - Chapters 8-9 Study Guide

## In-Class Exam (Chapter 8)

1. Write the linear system of differential equations in matrix form. Look at problems 8.1.1-6
2. Verify that the matrix X is a solution of the given system. Look at problems 8.1.11-16.
3. Find the general solution to the system. Show work! Look at problems 8.2.1-12.
4. Find the general solution to the system. Use Derive to find eigenvalues and eigenvectors. Two parts. Look at problems in section 8.2.
5. Find eigenvectors and the general solution to a system with repeated eigenvalues. The eigenvalues are given. Look at problems 8.2.19-28.
6. Use the method of undetermined coefficients to solve the given system. Look at problems 8.3.1-8.
7. Use variation of parameters to solve the given system. Look at problems 8.3.11-30.
8. 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 .
9. 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.
10. 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.

1. y' = xy + sqrt(y), y(0) = 1; y(0.5)
1. Create a table comparing Euler's method, Improved Euler's method, and the fourth-order Runge-Kutta method for h = 0.1.
2. Create a table comparing Euler's method, Improved Euler's method, and the fourth-order Runge-Kutta method for h = 0.05.
2. y' = x + y2, y(0) = 0; y(0.5)
1. Create a table comparing Euler's method, Improved Euler's method, and the fourth-order Runge-Kutta method for h = 0.1.
2. 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

 # Pts 1 2 3 4 5 6 7 8 9 10 TH Total 3 5 10 16 8 10 10 6 6 10 16 100