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Unformatted text preview: GE 207K Method of Undetermined Coefficients September 29, 2010 This page aims to recap our mini-lecture on solving 2nd order nonhomogeneous ODEs with constant coefficients using the method of undetermined coefficients. We are dealing with 2nd-order linear nonhomogeneous ordinary differential equations with constant coefficients which can be written as y 00 + ay + by = f ( t ) , (1) where a and b are two CONSTANTS , and f ( t ) is either a sine, cosine, exponential, or polynomial (or constant). In this method, we will wisely choose a form for the particular solution which will have some unknown coefficients. Later solve for these coefficients. Steps : 1. Find the corresponding homogenous solution, y h : Set the f ( t ) equal to zero, and find the corrensponding solution to the differential equation. In other words, solve the differential equation y 00 + ay + by = 0 (2) and call the solution y h . 2. Find the particular solution, y p : (i) For a given f ( t ) , choose a form for y p from table below: f ( t ) y p Ke t Ae t K cos t A cos t + B sin t K sin t Ke t cos t e t ( A cos t + B sin t ) Ke t sin t Kt 4 At 4 + Bt 3 + Ct 2 + Dt + E 1 , 5 , 10 , 31 , 59 , etc. A Kt 3 e t ( At 3 + Bt 2 + Ct + D ) e t (ii)...
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- Fall '10