where are all constants and .
For a linear non-homogeneous differential equation, the general solution is the superposition of the particular solution and the complementary solution .
The complementary solution which is the general solution of the associated homogeneous equation () is discussed in the section of Linear Homogeneous ODE with Constant Coefficients. This section summarizes common methodologies on solving the particular solution .
Method of Undetermined Coefficients: The non-homogeneous term in a linear non-homogeneous ODE sometimes contains only linear combinations or products 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. (See further detail.)
Method of Variation of Parameters: If the complementary solution has been found in a linear non-homogeneous ODE, one can use this complementary solution and vary the coefficients to unknown parameters to obtained the particular solutions. This methods is called the method of variation of parameters. (See further detail.)
Method of Reduction of Order: When solving a linear homogeneous ODE with constant coefficients, we factor the characteristic equation to obtained the homogeneous solution. Similarly, the method of reduction of order factors the differential operators and inverses (integrates) them one by one to reduce the order and eventually obtain the particular solution. (See further detail.)
Method of Inverse Operators: The method of inverse operators takes a step further than the method of reduction of order by categorizing how the inverse differential operator and its higher order operators affect common functions to achieve a more systematic way to obtain the particular solution. (See further detail.)