Finding equilibria and phase plane for dx/dt = y, dy/dt = -y² - sin(x)

[MrBioMedic]

Hello everyone I am hoping to get a little with a system.

Here is the the system:

dx = y
dt

dy = -y^2 - sin(x)
dt

I need to find all the equilibira and determine whether they are sinks, sources, etc...

I need to sketch the x and y nullclines.

Indicate the direction of the solution curve in any regions bounded by the nullcines.

Lastly, sketch the entire phase plane.



Thank you for the help in advance!

 

[arildno]

What have you done so far?

 

[HallsofIvy]

An "equilibrium" point is where the function is a constant: you must have
$$\frac{dx}{dt}= y= 0$$
and
$$/frac{dy}{dt}= -y^2- sin9x)= 0$$.

What does that tell you?