Differentiation - Equilibria The size z(t) of a hailstone evolves according to the differential equation $ \frac{dz}{dt}= A\sqrt(z) - B\sqrt(z^3) $ where A and B are positive constants. Without solving the differential equation, determine the limiting size $lim_{t→+∞} z(t) $in the case where $z(0)=1$ thinking in terms of equilibria. I have no idea of equilibria. Can anyone please explain this in detail?

An equilibrium for the autonomous differential equation $\dfrac{dz}{dt} = f(z)$ is a constant solution, i.e. a value of $z$ for which $f(z) = 0$. If you start out at an equilibrium, you stay there. In an interval not containing an equilibrium, the sign of $f(z)$ is constant (assuming $f$ is continuous). As $t \to \infty$, you either approach an equilibrium or go off to $\infty$, depending on that sign.

For example, if $z=1$ and $z=2$ were the equilibria and $f(z) < 0$ for $1 < z < 2$ while $f(z) > 0$ for $z > 2$, the limit would be $1$ if $1 < z(0) < 2$ or $\infty$ if $z(0) > 2$. If the signs were reversed, the limit would be $2$.

So what you want to do in your problem is

1. Find the equilibria
2. Determine the sign of $A \sqrt{z} - B \sqrt{z^3}$ in the intervals between equilibria, and between equilibria and $\infty$
3. Conclude from that information what the limit will be.