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 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

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

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