How would I go about sketching the x vs t and y vs t graphs for the following picture:
Our class didn't get the opportunity to solve these types of questions, so I'm on my own for this one. My thinking is that, for the first one at least, that it would need to be parametrized in some way because it looks like a circle. Since it's a circle of radius $4$, the differential equation must be related to the general formula of the circle. My problem is that I'm not sure where to even start since I'm not given any information about t, but I strongly believe that parametrization should play a key role here.
Any suggestions on how to initially proceed, at least for the first problem?
For the first one, try $$x' = y, y' = - x, x(0) = 0, y(0) = 4$$
This results in $$x(t) = 4 \sin t \\ y(t) = 4 \cos t$$
You should be able to draw those two solutions for $x(t)$ and $y(t)$.