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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?

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    $\begingroup$ These are not "equations", let alone ODEs, but function terms. You can find out about their behavior by looking at their graphs. $\endgroup$ Nov 19, 2012 at 9:41

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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.

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  • $\begingroup$ Can you explain why? i.e. What does having a term in $xe^x$ have to do with returning to equilibrium quickly, etc? $\endgroup$ Nov 19, 2012 at 2:20
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    $\begingroup$ @Imray: it means you have a double root in the characteristic equation, which puts it on the boundary between two real roots (overdamped) and two complex roots (underdamped) $\endgroup$ Nov 19, 2012 at 2:46
  • $\begingroup$ Can you give me a real world example of overdamped and critically damped? I understand that under-damped is the motion of an ordinary spring system or pendulum that dies down over time, but I can't picture the other two. $\endgroup$ Nov 19, 2012 at 2:54
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    $\begingroup$ @Imray: they are still springs or pendulums (pendula?) but with so much friction they don't overshoot. that is why critically damped approaches equilibrium fastest. Overdamped is like moving through molasses-you just can't get there very fast, so reducing the damping is a good thing. Underdamped gets you to equilibrium faster, but then you overshoot. $\endgroup$ Nov 19, 2012 at 3:02
  • $\begingroup$ Just did a lot of studying on the topic yesterday, I understand it now. Thank you Ross $\endgroup$ Dec 12, 2012 at 20:55
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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.

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