Applications


Part a: 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)(cos(2t)+sin(2t)) is an underdamped solution.
y=e^(-t)(cos(2t)+sin(2t)) 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)) 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=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.


Please enter any multiplication explicitly with * symbols so the parcer can understand your answer, like this:

2t3e5t2⟶2∗t3∗e(5∗t2).2t^3e^{5t^2} \longrightarrow 2*t^3*e^{(5*t^2)}.

Problem 1

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

y′′−2y′−3y=0y'' - 2y' - 3y = 0

Initial Conditions to Check: