Find fixed points for a system of non-linear differential equations

Question

I'm trying to find the fixed points for the following system:

$$\frac{dx}{dt} = 3x -y^2$$ $$\frac{dy}{dt} = \sin(y)-x$$

Both $\frac{dy}{dt}$ and $\frac{dx}{dt}$ equal zero at the fixed point(s). So I've gotten $y = \sqrt{3x}$ out of the first equation and plugged that into the second equation.

$$\sin(\sqrt{3x}) - x = 0$$

Now I'm stuck solving this equation for x. I've managed to calculate the zeros/roots using Matlab, giving me $x_1 = 0$ and $x_2 = 0.9866$ as solution.

Is there a way to find the zeros analytically? Thanks in advance!

Answer 1

Hint: i would write $$x=\frac{1}{3}y^2$$ plugging this in the second equation we get $$3\sin(y)-y^2=0$$ This equation has only two solutions: $$y=0$$ or $$y\approx 1.722125112$$

Answer 2

With regards to the numerical approach, take: $$3\sin(y)=y^2$$ Then $$y=\sqrt{3\sin(y)}$$Then we can say: $$y_{n+1}=\sqrt{3\sin(y_n)}$$ Input some $y_0$, say $1$, and iterate ($\text{Ans}$ key on a calculator will do that)