2^nd Order Linear Homogeneous Differential Equations

General solutions to

The characteristic equation for this differential equation is

. Let the solutions of this quadratic be m₁ and m₂.

We have three cases to consider:

1. Distinct Real Roots: m₁ and m₂ are real and m₁ ≠ m₂. Then the solutions to the differential equation are

2. Equal Real Roots: m₁ and m₂ are real and m₁ = m₂. Then the solutions to the differential equation are
   
   
3. Complex Roots: m₁ = A + Bi and m₂ = A - Bi. Then the solutions to the differential equation are