# Solving Second Order Differential Equations with Repeated Roots

fact
Remember that the "characteristic equation" for a linear second order differential equation $$ay'' + by' + cy = 0$$ is given by: $$ar^2 + br + c = 0$$
fact
A second order differential equation has repeated roots if the roots of its characteristic equation are both the same real number $$\lambda$$.
fact
For repeated roots the two solutions to the homogeneous equation are: $$y_1 = e^{\lambda t}$$ and $$y_2 = te^{\lambda t}$$ We'll cover why this is the case a little later, it's more confusing than helpful right now.
fact
The general solution is given by: $$y = c_1e^{\lambda t} + c_2te^{\lambda t}$$
practice problems