Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $e^{-t}(c_1cos(3t) +c_2sin(3t))$ is underdamped?

share|improve this question
These are not "equations", let alone ODEs, but function terms. You can find out about their behavior by looking at their graphs. –  Christian Blatter Nov 19 '12 at 9:41

2 Answers 2

up vote 2 down vote accepted

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.

share|improve this answer
Can you explain why? i.e. What does having a term in $xe^x$ have to do with returning to equilibrium quickly, etc? –  Imray Nov 19 '12 at 2:20
@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) –  Ross Millikan Nov 19 '12 at 2:46
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. –  Imray Nov 19 '12 at 2:54
@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. –  Ross Millikan Nov 19 '12 at 3:02
Just did a lot of studying on the topic yesterday, I understand it now. Thank you Ross –  Imray Dec 12 '12 at 20:55


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.

share|improve this answer
But my question was deeper. Why do repeated roots mean it's critically damped? Why do distinct roots mean its overdamped? –  Imray Nov 19 '12 at 15:36
hyperphysics.phy-astr.gsu.edu/hbase/oscda2.html take a look at that site. –  diimension Nov 20 '12 at 1:38

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.