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3.5 Method of Undetermined Coefficients

# 3.5 Method of Undetermined Coefficients - GE 207K Summary...

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GE 207K Method of Undetermined Coefficients October 3, 2011 Summary: We are dealing with 2nd-order linear nonhomogeneous ordinary differential equations with constant coefficients which can be written as y 00 + ay 0 + by = f ( t ) , (1) where a and b are two CONSTANTS , and f ( t ) is either a sine, cosine, exponential, or polynomial (or a 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 0 + 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 γt Ae γt K cos ωt A cos ωt + B sin ωt K sin ωt Ke αt cos ωt e αt ( A cos ωt + B sin ωt ) Ke αt sin ωt Kt 4 At 4 + Bt 3 + Ct 2 + Dt + E 1 , 5 , 10 , 31 , 59 , etc. A Kt 3 e γt ( At 3 + Bt 2 + Ct + D ) e γt (ii) Modification rule – If the form chosen for y p appears in y h , multiply your choice for y p by t , and recheck . (See Example 2 for an example of this). Keep multiplying y p by t

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