- You are given a 2×2 system of linear differential equations. Write the system in matrix form, find the characteristic equation, find the eigenvalues, find the eigenvectors, write the general solution, and the find the constants that satisfy the initial conditions.
- The solution to a 2×2 system of linear differential is given. List the eigenvalues, find the characteristic equation, and then find a square matrix \( \mathbf{A} \) so that \( \mathbf{X}' = \mathbf{A} \mathbf{X} \) has those eigenvalues.
- Given \( \left ( s\mathbf{I} - \mathbf{A} \right)^{-1} \) for a 2×2 system, find the matrix exponential. The expression has already had partial fractions applied to it, so this will save you time, but you will need to know the inverse Laplace transforms. Then, given the initial conditions, find the solution to the initial value problem.
- A 3×3 system \( \mathbf{X}' = \mathbf{A} \mathbf{X} \) is given in matrix form along with the graph of the characteristic polynomial. Find the fundamental matrix.
- A 3×3 system \( \mathbf{X}' = \mathbf{A} \mathbf{X} \) is given in system form along with the graph of the characteristic polynomial. Find the general solution.
- 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.
- For the non-homogeneous system \( \mathbf{X}' = \mathbf{A} \mathbf{X} + \mathbf{F}\), you are given \( \mathbf{X}(0) \), \( \mathbf{F} \), \( \mathbf{\Phi} \), and \( \mathbf{\Phi}^{-1} \). Find the solution.
- Classify the differential equation according to order, linearity, type of derivative, homogeneity, and the ASLEHBN acronym from chapter 4. 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.

- There should be room on the exam to work the questions, but you may want to bring scratch paper.
- #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, although the Excel file should be uploaded into Canvas ahead of time.
- You should know the big-4 of the common Laplace transforms.

# | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Total |
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Pts | 14 | 7 | 7 | 9 | 9 | 9 | 9 | 9 | 14 | 15 | 102 |