## Ordinary Differential and Difference Equations

### Week 2

##### Problem 3.1

(a) The conditions that allow for simple undamped harmonic motion are when there are no outside forces of friction or gravity being accounted for or affecting the system.

(b) See image below for the homogeneous solution to the problem. Note that the amplitude does not depend on frequency.

(c) Here is the particular solution to the inhomogeneous problem, followed by the plot for its magnitude and phase (plotted using numpy and matplotlib in Python).

(d) To find the Q factor, I know that it is the ratio of the center frequency to the width of the curve where the magnitude drops to half of its maximum, so looking at the plot from above zoomed in (see below), we can see that the center frequency is right around 1 rad/s, then the width of the curve is about 0.2 rad/s. This gives us a Q factor of 5. I was unable to get much farther with this solution as I floundered, unfortunately.

(e) I made it through using Laplace transforms quite far until I encountered a pretty complex fraction in the Laplace domain. I tried taming it, but am not sure that I got it right since it doesn't quite match with previous solutions obtained using other methods.

(f) I skipped this portion so I could focus on solving the next two problems since it wasn't as obvious as to how I'd approach it.

##### Problem 3.2

After reviewing some notes from my ODE class from undergrad, I was able to remember how to derive the equations of motion for the system, which was necessary to set up the system of equations for this problem and then simplify it. You can see those below. I floundered quite a bit determining how to put everything into a matrix so I could diagonalize it, but didn't figure it out.