Mathematics

Material on Test 3

Test 3 is Friday, April 22. The test will be on Secs. 9.1 - 9.7.

Calculators may be used for arithmetic only. You may also use the table of integrals in the front of the text. In addition, you may bring to the test one sheet of paper with whatever you find useful written on it.

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 and a fundamental matrix 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.

Some additional problems: Sec. 9.4, problem 27. Sec. 9.5, problems 19, 21. Sec. 9.6, problems 5, 7.

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