Applications


Damped Motion


We learned in class that there are three kinds of damped motion, based on the value of the descriminant D=b2−4acD = \sqrt{b^2 - 4ac}.


Underdamped

Here is an example of underdamped motion.

$y=e^{-t}\left( \cos(2t)+\sin(2t) \right)$ is an underdamped solution.
y=e−t(cos⁡(2t)+sin⁡(2t))y=e^{-t}\left( \cos(2t)+\sin(2t) \right) is an underdamped solution.

Overdamped

Here is an example of overdamped motion.

$y=e^{-3t}(\cos(2t)+\sin(2t))$ is an overdamped solution.
y=e−3t(cos⁡(2t)+sin⁡(2t))y=e^{-3t}(\cos(2t)+\sin(2t)) is an overdamped solution.

Critically Damped

Here is an example of underdamped motion.

$y=e^t(\cos(2t)+\sin(2t))$ is a critically damped solution.
y=et(cos⁡(2t)+sin⁡(2t))y=e^t(\cos(2t)+\sin(2t)) is a critically damped solution.

You will see three differential equations below. They are all either over-damped, under-damped, or critically damped.

Find a particular solution for each equation and classify your solution.

No initial conditions are given because you are only asked to find one solution.


Problem 1

Find a family of functions that solves the equation and satisfies the conditions below.

y′′−2y′−3y=0y'' - 2y' - 3y = 0