In mathematics, the method of undetermined coefficients is an approach to finding a particular solution to certain nonhomogeneous 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, an ansatz or '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.

Description of the method

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

:<math> \sum_{i=0}^n c_i y^{(i)} + y^{(n+1)} = g(x)</math>

:where <math>y^{(i)}</math> denotes the i-th derivative of <math>y</math>, and <math>c_i</math> denotes a function of <math>x</math>.

The method of undetermined coefficients provides a straightforward method of obtaining the solution to this ODE when two criteria are met:

  1. <math>c_i</math> are constants.
  2. g(x) is a constant, a polynomial function, exponential function <math>e^{\alpha x}</math>, sine or cosine functions <math>\sin{\beta x}</math> or <math>\cos{\beta x}</math>, or finite sums and products of these functions (<math>{\alpha}</math>, <math>{\beta}</math> constants).

The method consists of finding the general homogeneous solution <math>y_c</math> for the complementary linear homogeneous differential equation

:<math> \sum_{i=0}^n c_i y^{(i)} + y^{(n+1)} = 0,</math>

and a particular integral <math>y_p</math> of the linear non-homogeneous ordinary differential equation based on <math>g(x)</math>. Then the general solution <math>y</math> to the linear non-homogeneous ordinary differential equation would be

:<math>y = y_c + y_p.</math>

If <math>g(x)</math> consists of the sum of two functions <math>h(x) + w(x)</math> and we say that <math>y_{p_1}</math> is the solution based on <math>h(x)</math> and <math> y_{p_2}</math> the solution based on <math>w(x)</math>. Then, using a superposition principle, we can say that the particular integral <math>y_p</math> is