Mathematics

### Material on the Final Exam

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The final exam is Tuesday, May 8, 1:00 - 4:00 p.m., in the usual room.

Calculators may be used for arithmetic and function evaluations only.

Please bring your text to the final exam. You can use the information on the inside front and back covers. (You can bring a printout of these pages if that is more convenient.)

The exam will cover first-order equations (Secs. 1.1 - 1.3, 2.2 - 2.3, pp. 32 - 34 (pp. 34 - 36 in the 5th edition), and 3.2 only), second-order linear equations (Secs. 4.2 - 4.6 only), Laplace transforms (Secs. 7.2 - 7.6, 7.8), linear systems (Secs. 9.2 - 9.7), and the phase plane (Secs. 5.4 and 12.1 - 12.3).

For first-order equations, second-order linear equations, Laplace transforms, and linear systems, you can use the review sheets for Tests 1, 2 and 3. Omit material from sections that are not listed above.

Things you should be able to do from Secs. 5.4 and 12.1 - 12.3:

Sec. 5.4. For x'=ax+by, y'=cx+dy, sketch some direction vectors, use a sketch of direction vectors to sketch trajectories, solve the phase plane equation, use your solution to sketch trajectories. Don't forget to include the direction of trajectories.

Sec. 12.2. For x'=ax+by, y'=cx+dy, use eigenvalues and eigenvectors to sketch trajectories. Use eigenvalues to identify equilibria as attracting or repelling nodes, spiral attractors or repellers, saddles, or centers.

Secs. 12.1 and 12.3. For x'=f(x,y), y'=g(x,y), find nullclines, use to sketch typical vectors, find equilibria, find the matrix of the linearization at each equilibrium, find its eigenvalues and eigenvectors, use this information to help sketch the trajectories of the nonlinear system.

I suggest reviewing Tests 1, 2 and 3, as well as the practice final.