StudyGuide_Mid2.pdf - MATH 4BDifferential Equations Fall...

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MATH 4B–Differential Equations, Fall 2016 Midterm 2 Study Guide GENERAL INFORMATION AND MIDTERM RULES The exam will have a duration of 50 minutes. No extra time will be given. Failing to submit your solutions within 50 minutes will result in your exam not being graded. The material included for the first midterm corresponds to everything we have covered until Monday, November 14th. The sections are 3.1–3.7, and 6.1–6.4. You can bring ONE index card of dimensions up to 4 00 × 5 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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LIST OF SKILLS Decide whether two functions y 1 ( t ) , y 2 ( t ) form a fundamental set of solutions for a second order linear ODE, i.e., are linearly independent. Find the general solution for a second order homogeneous ODE with constant coeffi- 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 coefficients using the method of undetermined coefficients. Find a particular solution for a second order ODE with constant coefficients using the method of variation of parameters. Find the general solution to a second order ODE with constant coefficients. Solve Initial Value Problems associated to a second order ODE with constant coeffi- cients. Find and solve a differential equation modeling the position u ( t ) of a spring. Compute the Laplace transform of a given function. Compute inverse Laplace transforms of rational functions and piecewise continuous functions. Use Laplace transforms to solve second order IVP. 2
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SUMMARY A second order linear ODE is a differential equation of the form a ( t ) y 00 + b ( t ) y 0 + c ( t ) y = d ( t ) (*) The associated homogeneous equation is a ( t ) y 00 + b ( t ) y 0 + c ( t ) y = 0 ( ? ) Two solutions y 1 ( t ) and y 2 ( t ) for ( ? ) are said to form a fundamental set of solutions if they are linearly independent, i.e., if the Wronskian W ( y 1 , y 2 )( t ) = y 1 (
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