Method of Undetermined Coefficients

For a linear non-homogeneous ordinary differential equation with constant coefficients

where are all constants and , the non-homogeneous term sometimes contains only linear combinations or multiples of some simple functions whose derivatives are more predictable or well known. By understanding these simple functions and their derivatives, we can guess the trial solution with undetermined coefficients, plug into the equation, and then solve for the unknown coefficients to obtain the particular solution. This method is called the method of undetermined coefficients.

Trial Functions in the Method of Undetermined Coefficients: Some special cases and their trial solutions are listed as follows:

Format of
Non-homogeneous Term
Trial Function for Particular Solution




PS. Part (2) are basically multiples of two or more items in part (1).

Important! The above table holds only when NO term in the trial function shows up in the complementary solution. If any term in the trial function does appear in the complementary solution, the trial function should be multiplied by to make the particular solution linearly independent from the complementary solution. If the modified trial function still has common terms with the complementary solution, another must be multiplied until no common term exists.

Pros and Cons of the Method of Undetermined Coefficients:The method is very easy to perform. However, the limitation of the method of undetermined coefficients is that the non-homogeneous term can only contain simple functions such as , , , and so the trial function can be effectively guessed.