Example

Solve

Solution

This is a nonhomogeneous Cauchy-Euler equation as it has the form

Begin by finding the homogeneous solution. That is, solve

For Cauchy-Euler equations we guess a solution of

So:

and

Putting these values into Equation (1) give us

Factoring out the $x^ m$ term leaves us with

This equation has two distinct real roots, so the homogeneous solution has the form

Next we need to find a particular solution, $y_ p(x)$. We can do this using the method of Variation of Parameters. A particular solution is given by

where $y_1 = x^{-1}$ and $y_2 = x^{3}$ To find $r(x)$ we need to make the coefficient of $y''$ equal to 1 in the original equation by dividing through by $x^2$. That is

So $r(x) = 2x$.

The general solution is given by the sum of the homogeneous solution and the particular solution: