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Differential Equations and Orthogonal Decompositions



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AMTH251a Differential Equations and Orthogonal Decompositions Fall 2005 Yale University Lectures MWF 10 30 11 20 BCT 508 Instructor Amit Singer amit singer yale edu Room 103 in Watson Bldg Phone 2 4345 Office hours Th 4 5PM Text Erwin Kreyszig Advanced Engineering Mathematics 8th Edition John Wiley Sons 1999 Supplementary material to be distributed In this class I will cover ordinary differential equations and systems 7 weeks Fourier series and orthogonal functions 4 weeks and applications of the Fourier eigenfunction method to boundary value problems and partial differential equations 2 weeks Emphasis is on the mathematical techniques and concepts that are important in the engineering and physical sciences By taking this course a student will acquire a solid mathematical knowledge base for junior senior classes in the applied sciences as well as a proper intuition for later more theoretical courses such as AMTH260 where rigorous proofs are constructed Prerequisites for the course are multivariate calculus and linear algebra the latter could be taken concurrently A natural continuation of the class is AMTH252b in the spring term which will include transforms variational calculus and more on partial differential equations Course Work and Grading Policy There will be weekly homework assignments and a final exam The final grade will be based 30 on the homework and 70 on the final exam 1 Course Schedule Tentative First order differential equations Week of 8 31 Week of 9 5 Week of 9 12 Differential equations Basic concepts and ideas Geometrical meaning of y 0 f x y Direction fields and solution curves Linear 1st order equations Applications Separable equations Differences between linear and non linear equations Existence and uniqueness theorem Domain of validity of a solution Substitution methods and exact equations More applications e g population models stability Review of linear algebra and complex numbers Linear Differential Equations of Higher Order Week of 9 19



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