The Edge Case
If the characteristic equation has a repeated root \(r_1 = r_2 = r\), we need a second independent solution. We find it by multiplying by \(x\):
\[y = C_1 e^{rx} + C_2 x e^{rx}\]
This state is called Critical Damping. It is the fastest way for a vibrating system to return to rest without overshooting.
Worked Examples
Example 1: Repeated Root
Solve \(y'' + 6y' + 9y = 0\).
- Eq: \(r^2 + 6r + 9 = 0 \implies (r+3)^2 = 0\).
- Root: \(r = -3\) (repeated).
- Result: \(y = C_1 e^{-3x} + C_2 x e^{-3x}\).
The Bridge to Quantum Mechanics
Critical damping is a classical concept, but the "repeated root" math appears in Quantum Mechanics during Degeneracy. Sometimes, two different physical states of an atom have exactly the same energy. When this happens, the differential equations for those states are "repeated." We use the math of linear independence (like adding the \(x\) factor) to ensure we can distinguish between these states, allowing us to accurately count the number of electrons an atom can hold in each shell.