Question

The image below is a direction field for the differential equation y'=xsiny.
(i)y(0)=1 (ii) y(0)=2 (iii)y(0)=pi
(iv)y(0)=4 (v) y(0)=5
The graph makes sense.
(b) Find all the equilibrium solutions.
y(0)=pi is the equilibrium solution, but how were they able to determine that y=n(pi) describes all the equilibrium solutions?

Answer 1

That's the solution to (b)^^^^^^^
Just trying to understand it.

Answer 2

For all integers n, sin(nπ)=0. So x*sin(nπ)=y'=0. Also, y'=0 if x=0. But for any small change in x, y' no longer equals zero unless y=nπ.

Answer 3

thanks @Herp_Derp