Solving Second Order Differential Equations with Repeated Roots

Remember that the "characteristic equation" for a linear second order differential equation \(ay'' + by' + cy = 0\) is given by: $$ ar^2 + br + c = 0 $$

A second order differential equation has repeated roots if the roots of its characteristic equation are both the same real number \(\lambda\).

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.

The general solution is given by: $$ y = c_1e^{\lambda t} + c_2te^{\lambda t} $$