StudyGuide_FinalExam - MATH 4BDierential Equations Fall...

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MATH 4B–Di erential Equations, Fall 2016 Final Exam Study Guide GENERAL INFORMATION AND FINAL EXAM RULES The exam will have a duration of 3 hours. No extra time will be given. Failing to submit your solutions within 3 hours will result in your exam not being graded. The Final Exam is comprehensive. The sections are 1.1–1.3, 2.1–2.6, 3.1–3.7, 6.1–6.4, 7.1–7.9. 35% of the questions will be from Chapters 1 and 2, 35% from Chapters 3 and 6, and 30% from Chapter 7. You can bring ONE index card of dimensions up to 5 00 6 00 . This index card should be handwritten and can be filled on both sides. However, note cards of higher dimensions than the ones mentioned above or typewritten WILL NOT be allowed. Calculators WILL NOT be needed, nor allowed for this exam. Last but not least, CHEATING WILL NOT BE TOLERATED. 1
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SKILL’S LIST Verify that a given function is a solution for an Initial Value Problem (IVP). Sketch the direction (slope) fields of an ODE. Find equilibrium solutions for Au- tonomous ODEs, and determine whether equilibrium solutions are semistable or not. Solve a separable ODE and corresponding IVP. Find the general solution to a first order linear ODE. Determine if an ODE is exact. Solve an exact ODE. Read information from a word problem, and establish the corresponding ODE modeling the situation in the following cases: 1. Free falling object. 2. Population growth and decay. 3. Tank model. 4. Newton’s law of cooling. 5. Springs. Determine whether two functions y 1 ( t ) , y 2 ( t ) form a fundamental set of solutions for a second order linear ODE. Find the general solution for a second order homogeneous ODE with constant coe ffi - cients. Given a solution y 1 ( t ) for a second order linear ODE, use reduction of order to find a second (independent) solution of the form y 2 ( t ) = u ( t ) y 1 ( t ). Find a particular solution for a second order ODE with constant coe ffi cients using the method of undetermined coe ffi cients, and/or variation of parameters. Find the general solution to a second order ODE with constant coe ffi cients. Solve Initial Value Problems associated to a second order ODE with constant coe ffi - cients. 2
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Compute the Laplace transform of a given function. Compute inverse Laplace transforms to rational functions and piecewise continuous functions. Use Laplace transforms to solve second order IVP. Compute the determinant of a matrix (2 2 or 3 3 would su ffi ce). Find the eigenvalues and eigenvectors of a matrix. Find the inverse of a matrix. Find the canonical Jordan form of a 2 2 matrix. Find the general solution of a homogeneous system of linear ODEs with constant coe ffi cients. Find the fundamental matrix of a system of linear ODEs with constant coe ffi cients.
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