Looking at an equation, how can you know if if it overdamped, critically damped, or under-damped?
For example:
How can you tell that the equation $c_1e^{2x} + c_2e^{-2x}$ is overdamped?
How can you tell that the equation $e^{-x}(c_1+c_2x)$ is critically damped?
How can you tell that the equation $e^{-t}(c_1\cos(3t) +c_2\sin(3t))$ is underdamped?
Best Answer
The shape of these is the key. An overdamped system will be pure exponentials (though they are usually all decreasing). Critically damped has a term in $xe^x$. And underdamped have oscillatory solutions, like yours with cosine and sine waves.