The Variation of Parameters Formula

Variation of Parameters is a method for computing a particular solution
to the nonhomogeneous linear second-order ode:

Solution Procedure

There are two steps in the solution procedure:

(1) Find two linearly independent solutions to the homogeneous problem

Call these solutions y1(t) and y2(t).

(2) Use the variation of parameters formula to determine the particular solution:

where W(t), called the Wronskian, is defined by

According to the theory of second-order ode, the Wronskian is
guaranteed to be non-zero, if y1(t) and y2(t) are linearly independent.

Example

Consider, for example, the ode

The homogeneous equation is

Two linearly-independent solutions to the ode are:

The Wronskian is:

The particular solution is

The integrals can be determined using integration by parts. The first integral is

The second integral is

Putting everything together, we have:

The general solution to the differential equation is

If you have questions or comments, don't hestitate to contact us.