If I know a function satisfies a first order ODE, can I find an expression for its second derivative?

Question

Suppose I have a differential equation

$$ p^\prime = p(1-p) $$

Is there way to find a second order differential equation for $p$?

$$p'' = F(p)$$

I suspect it has to do with clever use of the chain rule

Answer 1

You can differentiate your ODE and substitute in your expression for $p'$ as a function of $p$.

Answer 2

I get this expression

$$p''=2p^3-3p^2+p$$

Answer 3

Just take the derivative of p' : $$p^\prime = p(1-p)$$ $$p''=p'-2pp'=p'(1-2p)=p(1-p)(1-2p)$$