Consider the DE
which is linear with constant coefficients.

First we will work on solving the corresponding homogeneous equation. The auxiliary equation (using m as your variable) is which has root .

Because this is a repeated root, we don't have much choice but to use the exponential function corresponding to this root:
to do reduction of order.
.
Then (using the prime notation for the derivatives)
So, plugging into the left side of the differential equation, and reducing, we get

So now our equation is .
To solve for u we need only integrate twice, using a as our first constant of integration and b as the second we get
Therefore , the general solution.

We knew from the beginning that was a solution. We have worked out is that is another solution to the homogeneous equation, which is generally the case when we have multiple roots. Then is the particular solution to the nonhomogeneous equation, and the general solution we derived is pieced together using superposition.