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ODE Lecture Notes Section 3.5 Page 1 of 5 Section 3.5: Nonhomogeneous Equations; Method of Undetermined Coefficients Big idea: One way to solve a nonhomogeneous linear second-order differential equations with constant coefficients ay by cy g t is to “guess” a particular solution that has the same form as g(t), then work out the value of the coefficient that makes that guess work. Big skill: You should be able to solve nonhomogeneous second order differential equations with constant coefficients using the method of “undetermined coefficients,” which is to say that you should be able to find the homogeneous solution, the particular solution and its coefficient, and then add them together for the general solution. The general solution to the nonhomogeneous equation will be the general solution of the corresponding homogeneous equation plus one or more additional terms that called the particular solution. This is because if yh(t) is the solution to ay by cy 0 , and yp(t) is the solution to ay by cy g t , then y t yh t y p t is the most general solution to the nonhomogeneous equation because ay t by t cy t ayh t byh t cyh t ay t by p t cy p t p 0 g t g t Theorem 3.5.1: The Difference of Nonhomogeneous Solutions Is the Homogeneous Solution If Y1 and Y2 are two solutions of the nonhomogeneous equation L y y p t y q t y g t , then their difference Y1 Y2 is a solution of the corresponding homogeneous solution L y y p t y q t y 0 . If, in addition, y1 and y2 are a fundamental set of solutions of the homogeneous equation, then Y1 Y2 c1 y1 t c2 y2 t , for certain constants c1 and c2. Theorem 3.5.2: General Solution of a Nonhomogeneous Equation The general solution of the nonhomogeneous equation L y y p t y q t y g t , can be written in the form y t c1 y1 t c2 y2 t Y t m where , y1 and y2 are a fundamental set of solutions of the corresponding homogeneous equation, c1 and c2, are arbitrary constants, and Y is some specific solution of the nonhomogeneous equation. Method of Undetermined Coefficients: When g(t) is an exponential, sinusoid, or polynomial, make a guess for the particular solution that is the same exponential, sinusoid, or polynomial, except with arbitrary coefficients that you determine by substituting your guess into the nonhomogeneous equation. ODE Lecture Notes Section 3.5 Page 2 of 5 Practice: 1. Solve the IVP y t y t 2 y t 2e3t , y(0) = 1, y’(0) = 1. ODE Lecture Notes Section 3.5 Page 3 of 5 2. Solve the IVP y t y t 2 y t 2sin t , y(0) = 1, y’(0) = 1. ODE Lecture Notes Section 3.5 Page 4 of 5 3. Solve the IVP y t y t 2 y t 2et sin t , y(0) = 1, y’(0) = 1. ODE Lecture Notes Section 3.5 Page 5 of 5 4. Solve the IVP y t y t 2 y t 2et , y(0) = 1, y’(0) = 1.

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posted: | 3/23/2011 |

language: | English |

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