Mathematics

Material on Test 3

Test 3 is Thursday, April 19. The test will be on Secs. 9.1 - 9.7 and 5.4.

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 reduced 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. 513 (p. 519 in the 5th edition).

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 a 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.

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

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