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Unformatted text preview: 3.5: Nonhomogeneous Equations and Undetermined Coefficients Recall Suppose a n y H n L + a n- 1 y H n- 1 L + + a 1 y ' + a y = f H x L has a general solution y H x L = y p H x L + y c H x L where the complementary solution y c is the general solution to the homogeneous ODE a n y H n L + a n- 1 y H n- 1 L + + a 1 y ' + a y = and y p is any particular solution to the nonhomogeneous ODE. Goal: Find a Particular Solution to the Nonhomogeneous ODE. Method of Undetermined Coefficients If f H x L is a linear combination of finite products of functions: A) A polynomial in x B) An exponential function e k x C) cos H k x L or sin H k x L Then, 1) Find the complementary solution y c 2) Construct a particular solution y p : A) If f H x L contains a term x m e x , where m is the largest such power, Then include H k + k 1 x + + k m x m L e x . Note that can equal zero! B) If f H x L contains a term x m e a x cos H b x L or x m e a x sin H b x L where m is the largest such power,...
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- Spring '11