How do you know if an equation of spring motion is overdamped?

Question

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?

Answer 1

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.

Answer 2

Hint:

Just look at how your equations are set up.

• $e^{-x}(c_1+c_2x)$ means you have repeated roots.
• $c_1e^{2x} + c_2e^{-2x}$ means you have distinct roots.
• $e^{-t}(c_1cos(3t) +c_2sin(3t))$ means you have complex conjugates roots.

The roots will tell you whether it is critically damped, overdamped, or underdamped.