Mathematics

Material on Test 3

Test 3 is Thursday, November 30. The test will be on Secs. 9.1 - 9.7, 5.4, and 12.2.

Calculators may be used for arithmetic only. You may also use the table of integrals in the front of the text.

Things you should be able to do:

Sec. 9.2. Use row-reduction to row-echelon form to find all solutions of a system of linear equations.

Sec. 9.3. Add matrices, multiply them by a constant, multiply matrices. Find the inverse of a square matrix using row-reduction. (For 2 by 2 matrices you may use the simple formula.) Find the determinant of a square matrix. Differentiate and integrate vector-valued and matrix-valued functions. Check that a vector or matrix function of t satisfies a differential equation. In addition, be sure you understand the theorem on p. 519.

Sec. 9.4. Rewrite a system of scalar differential equations as a differential equation in matrix form and vice-versa. Check that a collection of solutions of x'=Ax is a fundamental set of solutions, and use these solutions to give the general solution and a fundamental matrix. Also, use these solutions and a particular solution of x'=Ax+f to give the general solution of x'=Ax+f.

Secs. 9.5 and 9.6. Find eigenvalues and eigenvectors of a square matrix A. Use them to give the general solution for x'=Ax. Use the general solution or the fundamental matrix to solve initial value problems.

Sec. 9.7. Use the variation of parameters formula to find the general solution of x'=Ax+f and to solve initial value problems.

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 "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.

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