Preparation for Final Exam Read this pdf file from the coordinator Dr. Loo     Exam-1 and Exam-2 Materials: ü     Make sure you know how to classify any differential equations into linear or nonlinear and you can determine its order. ü     Make sure you understand the existence and uniqueness theorem TH2 page 285 and TH2 page 297. ü     Make sure you know how to solve any first differential equation by using (linear, separable, homog, Bernolli, exact, ....). ü     Make sure you know how to find the reduced-echelon form ü     Make sure you know how to show that a set of functions is linearly independent by Wronskian. ü     Make sure you know how to show that a set of vectors  is linearly independent. ü     Make sure you know how to show that a set of vectors  is a spanning set. ü     practice problems( 26 page41 seprable, 39 page 72 exact, 15 page54 linear, 15 page71 homog, 27 page71 Bernoulli) after solving the problem check your answers with answers to selected problems in the back of the textbook.     Materials After Exam-2: ü     Make sure you know how to find the a particular solution using variation of parameters ü     Make sure you know how to find the a particular solution using undeterminant coefficients ü     Make sure you know how to convert any system of differential equations into a first order system. ü     Make sure you find diagonal matrix D and invertible matrix P such that A=P D inv(P). and compute A^k = P D^k inv(P) ü     Make sure you know how to compute the characteristic equation and eigenvalues and eigenvectors for any 3x3 matrix (see an example) a common mistake is that you open all brackets then you can not factor the the characteristic equation. ü     Make sure you can state Cayley Hamilton theorem and use its applications. ü     Make sure you can solve any homog linear system of first order differential equaions X' = A X. (all cases distinct real eigenvalues, complex eigenvalues, repeated real eigenvalues, complete case ...) ü     Read Example 1 page 416, Example 3 page 422, Example 1 page 442 then solve the follwing problems: 24, 26 page 425 and 7, 8, 9 page 457 .after solving the problem check your answers with answers to selected problems in the back of the textbook.