I am attempting the following homework question:
Consider the autonomous ODE
$$
\frac{dy}{dx}=3y^{\frac{2}{3}}
$$
(a) Find the equilibrium values and determine if they are stable or unstable.
For this, I concluded that $y=0$ is the only equilibrium value, and that it is unstable, since $\frac{dy}{dx}>0$ for $y<0$ and $\frac{dy}{dx}>0$ for $y>0$.
(b) Try to solve the ODE using the method of separation of variables.
This is quite straightforward, and yields the solution $y=(x+c)^3$ for some $c \in \mathbb{R}$
(c) What solution does the method of separation of variables give, with $y(0)=0$?
Again, this is very straightforward, and the initial condition $y(0)=0$ implies that $y=x^3$ is the solution.
(d) Conclude that the IVP in the previous part does not have a unique solution.
Of course $y=0$ is also a solution, but am I missing something? This solution seems too trivial to be correct, especially given what the next part is asking for.
(e) Can you find other solutions?
I am at a loss for this one. Given that the previous parts were all about solving by the method of separation of variables, I get the impression that I should try and solve it using another method? How could I find more solutions?
