Mathematics

Material on the Final Exam

The final exam is Monday, May 9, 1:00 - 4:00 p.m., in the usual room.

Calculators may be used for arithmetic only.

Please bring your text to the final exam. You can use the integral table inside the front cover, and the Laplace transform table inside the back cover.

You may bring to the test four sheets of paper with whatever you find useful written on them. I suggest reusing the sheets you made for Tests 1, 2, and 3, plus one sheet on later material.

The exam will cover first-order equations (Secs. 1.1 - 1.4 and 2.1 - 2.3 only), second-order linear equations (Secs. 4.1 - 4.5 only), Laplace transforms (Secs. 7.1 - 7.6), linear systems (Secs. 9.1 - 9.7), phase line and phase plane (pp. 34 - 36, Secs. 5.4 and 12.2 - 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 regarding the phase line and phase plane:

Pp. 34 - 36. Sketch the phase line, tell whether equilibria are attractors, repellers, or neither (i.e., attract from one side, repel from the other). The text uses a more confusing terminology. Use the phase line to predict asymptotic behavior of solutions.

Sec. 5.4. For x'=f(x,y), y'=g(x,y), find equilibria, sketch some direction vectors, use a sketch of direction vectors to sketch trajectories, solve the "xy 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.

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

I suggest reviewing Tests 1, 2 and 3. On Test 1, omit question 4; on Test 2, omit question 3.

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