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# Final Exam Review math 3113

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This 10 page Study Guide was uploaded by Marshall Essien on Tuesday May 10, 2016. The Study Guide belongs to math 3113 at University of Oklahoma taught by Dr. Sean Crowell in Spring 2016. Since its upload, it has received 50 views. For similar materials see intro to ordinary differential equations in Math at University of Oklahoma.

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Date Created: 05/10/16

MATH 3113.007 - Intro to ODE - FA14 Final Exam Form A NAME: Instructions: Complete all problems, showing your work in a clear and easy to follow manner. Please circle your answer. 1. Find the general solution of the following first order differential equations. (a) ???? + 2???????? = ???? ▯▯▯. ▯▯ ▯ ▯ (b) ▯▯= 4???? ???? ▯▯ ▯ ▯ (c) ▯▯ = ▯ 2 ▯ (HINT: the substitution ???? = ????/???? may be useful) 2. Consider the nonhomogeneous second order linear equation with constant coefficients given by ???? + 2???? + 10???? = ????, which models a forced spring-mass-dashpot system for an object with mass 1kg, damping constant 2kg per second, and spring constant of 10 N/m. (a) Find the function that describes the unforced motion of the object (i.e. the general solution of the associated homogeneous equation). (b) Find a particular solution of the forced equation (i.e the nonhomogeneous equation) using the Method of Undetermined Coefficients. (HINT: ???? is a linear polynomial of the form 1???? + 0) (c) What is the general solution of the nonhomogeneous equation, which describes the motion of the forced oscillator? 3. Use the (integral) definition of the Laplace transform (i.e not the table) to show ▯ directly that ℒ 2 = ▯. Make sure to use a limit to compute the improper integral! 4. Solve the initial value problem ???? + 4???? + 5???? = ????(???? − 1), ???? 0 = 0, ???? 0 = 0, ▯ using the Laplace transform. Show all of your work (don’t leave steps out) for full credit. 5. Consider the initial value problem given by the homogenous system ???? ???? ▯ 5 −1 ????▯ = ???????? ????▯ −1 5 ????▯ and accompanying initial condition ????▯0 = 1,???? 0▯= 3. Use the eigenvalue method to show that the solution to the initial value problem is given by ????▯ 2???? ▯▯− ????▯▯ ???? = ▯▯ ▯▯ . ▯ 2???? + ???? (Show all work for full credit, and use back for additional space if needed) MATH 3113.007 - Intro to ODE - FA14 Final Exam Form B NAME: Instructions: Complete all problems, showing your work in a clear and easy to follow manner. Please circle your answer. 1. Find the general solution of the following first order differential equations. ▯ ▯ ▯▯ ▯ (a) ???? + 3???? ???? = ???? . (b)▯▯ = 3???? ????▯ ▯ ▯▯ ▯▯ ▯ ▯ (c) ▯▯ = 3 ▯ ▯ (HINT: the substitution ???? = ????/???? may be useful) 2. Consider the nonhomogeneous second order linear equation with constant coefficients given by ???? + 4???? + 5???? = ????, which models a forced spring-mass-dashpot system for an object with mass 1kg, damping constant 2kg per second, and spring constant of 10 N/m. (a) Find the function that describes the unforced motion of the object (i.e. the general solution of the associated homogeneous equation). (b) Find a particular solution of the forced equation (i.e the nonhomogeneous equation) using the Method of Undetermined Coefficients. (HINT: ???? is a linear polynomial of the form 1???? + 0) (c) What is the general solution of the nonhomogeneous equation, which describes the motion of the forced oscillator? 3. Use the (integral) definition of the Laplace transform (i.e not the table) to show ▯ directly that ℒ 3 = ▯. Make sure to use a limit to compute the improper integral! 4. Solve the initial value problem ???? + 6???? + 10???? = ????(???? − 2), ???? 0 = 0, ???? 0 = 0, ▯ using the Laplace transform. Show all of your work (don’t leave steps out) for full credit. 5. Consider the initial value problem given by the homogenous system ???? ???? ▯ 3 −1 ????▯ = ???????? ????▯ −1 3 ????▯ and accompanying initial conditions ????▯0 = 3,???? 0▯= −1. Use the eigenvalue method to show that the solution to the initial value problem is given by ????▯ ???? ▯▯+ 2????▯▯ ???? = ▯▯ ▯▯ . ▯ ???? − 2???? (Show all work for full credit, use back for additional space if needed)

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