When trying to solve second order linear differential equations by using characteristic equations you may end up with repeated roots. In which chase you can use this method to find another root.
Given the following initial value problem:
we use a characteristic equation to find the roots:
We know that at least one of the solutions must be included in our general solution so lets write that one:
where our general solution is of the form:
due to some math magic we find that we can set the second solution to be a function of our first solution:
Don't ask me why.
We assemble this into our general solution:
and are now free to find the constants and solve the initial value problem.
See 1-Second Order Scalar DE (Sect 3-1 to 3-5) and Elementary Differential Equations - Kohler & Johnson - Second Edition - Pg. 127 for sources and more information.