Once you know how to solve second order linear homogeneous differential equations with constant coefficients, real or complex, the next step is to solve with those that have repeated roots. When solving for repeated roots, you could either factor the polynomial or use the quadratic equation, if the solution has a repeated root it means that the two solutions for “x” or whatever variable are the same.
Theorem for Solving Repeated Roots
Let: ay” + by’ + cy = 0
Be a differential equation such that the characterstic equation has the repeated root “r” That is :
(b^2)-4ac = 0
Then the general solution to the differential equation is given by
y = c1ert + c2tert