Method of undetermined coefficients

In mathematics, the method of undetermined coefficients is an approach to finding a particular solution to certain inhomogeneous ordinary differential equations and recurrence relations. It is closely related to the annihilator method, but instead of using a particular kind of differential operator (the annihilator) in order to find the best possible form of the particular solution, a "guess" is made as to the appropriate form, which is then tested by differentiating the resulting equation. For complex equations, the annihilator method or variation of parameters is less time consuming to perform.

Undetermined coefficients is not as general a method as variation of parameters, since it only works for differential equations that follow certain forms.

Consider a linear non-homogeneous ordinary differential equation of the form

The method consists of finding the general homogeneous solution ${\displaystyle y_{c}}$ for the complementary linear homogeneous differential equation

and a particular integral ${\displaystyle y_{p}}$ of the linear non-homogeneous ordinary differential equation based on ${\displaystyle g(x)}$. Then the general solution ${\displaystyle y}$ to the linear non-homogeneous ordinary differential equation would be

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