review problems for 2st midterm

review problems for 2st midterm - REVIEW INFORMATION AND...

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REVIEW INFORMATION AND PROBLEMS FOR THE SECOND EXAM. The second exam will cover: (a) qualitative phase plane methods for differential equations; text, sections 7.1—7.5; (b) Inner products; orthogonal and orthonormal bases; expansion in orthogonal bases; projec- tion on a span of orthogonal vectors and magnitude of error, Gram-Schmidt; sections 9.9, specifically 9.9.3, 9.10, and class notes (see lecture 17 on the syllabus page of the course web site). (c) Fourier series; text, 17.1—17.4, and 17.6. The formula sheet gives you the formula for the eigenvalues of a 2 × 2 matrix. You are expected to know how to translate this eigenvalue information into a classification of the singular point of a linear or affine differential equation in the plane, and, if it pertains to the linearization of a non- linear system at a singular point, how to use it to obtain the stability or type of the singular point. The formula sheet gives you orthogonal bases for various function spaces. You should know how to translate this information into the forumulas for the various type of Fourier series studied in chapter 17. Given a function f and an orthogonal basis { φ 1 , φ 2 , . . . } , the expansion of f with respect to this basis is X 1 h f, φ n i h φ n , φ n i φ n ( x ) .
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