Method of Variation of Parameters

Consider a linear non-homogeneous ordinary differential equation

where . The complementary solution of this equation can be written as

The method of variation of parameters finds the particular solution from varying the parameters of the complementary solution

where are unknown functions of to be determined.

To obtain n unknown functions , n equations are needed. One of them is the original differential equation which the solution must satisfy and the other n-1 can be imposed at our convenience.

For , its first derivative can be written as

Suppose we let , the complexity of is cut in half to

Repeat this step, let , and to the order, we have

Solve these n equations for and then integrate these first order derivatives back to . The integration constants can be neglected for only one particular solution is needed.

Pros and Cons of the Method of Variation of Parameters: The method of variation of parameters can also be used in linear differential equations with variable coefficients. However, the complementary solution must be found first and sometimes the final solution can not be obtained without numerical integration.