We solved the damped harmonic oscillator equation:
We chose a solution (ansatz) of the form
and computed the roots of the characteristic equation:
We found the roots to be:
We found that when , the roots are complex:
This means that the solution is oscillatory:
The solution is a damped oscillation with frequency .
When , the roots are real:
This means that the solution is not oscillatory:
where and .
The solution is the sum of two exponentials with different decay rates.
When , the roots are real and equal (repeated roots):
This means that the solution is not oscillatory, but also that our ansatz is not sufficient. The correct form of the solution is:
In most cases, we will work with weak damping.