Reduction of order

The reduction of order technique allows one to find a second solution to a differential equation when one solution is already found.

Take a linear second order constant coefficient homogeneous ODE (see Second order linear ODE) with one repeated-root solution. Add u(t) as follows

$y(t)=u(t)e^{2t}$

Substituting into the ODE, and you get

$u''=0$

Integrating up, and you get u in the form

$u=te^{2t}$