Method of Variation of Parameters

Consider the equation

In order to use the method of variation of parameters we need to know that {y₁, y₂} is a set of fundamental solutions of the associated homogeneous equation . We know that, in this case, the general solution of the associated homogeneous equation is y_h = c₁y₁ + c₂y₂. The idea behind the method of variation of parameters is to look for a particular solution such as

where u₁ and u₂ are functions. From this, the method got its name. The functions u₁ and u₂ are solutions to the system

which implies

where W(y₁, y₂) is the wronskian of y₁ and y₂. Therefore, we have

Example

Given that y₁ = x² and y₂ = x² ln x are solutions to x²y'' - 3xy' + 4y = x²ln x to the corresponding homogeneous differential equation, find the general solution to the nonhomogeneous differential equation.

First, we divide by x² to get the differential equation in standard form

y'' = 3/x y' + 4/x² y = ln x

We let

y_p = u₁y₁ + u₂y₂

The Wronskian matrix is

We use the adjoint formula to find the inverse matrix. First the Wronskian is the determinant which is

W = x³ + 2x³ ln x - 2x³ ln x = x³

So the inverse is

We have

Integrating by parts (with some effort) gives

u₁ = -(ln x)³/3

u₂ = (ln x)²/2

We have

y_p = -1/3 x²(ln x)³ + 1/2 x² (ln x)³ = 1/6 (ln x)³

Finally we get

y = c₁ x² + c₂ x² ln x + 1/6 (ln x)³