# Getting ALL solutions of the logistic ODE with DSolve?

In Mathematica 10.0, the command

DSolve[y'[x] == y[x]*(1 - y[x]), y[x], x]


gives

{{y[x] -> E^x/(E^x + E^C[1])}}


I take it the constant C[1] is assumed to be an arbitrary complex number, but I'm still missing the two solutions y[x]->0 and y[x]->1. Is there a way to force Mathematica to produce all solutions?

• You can take the limit C[1]->Infinity and C[1]->0, so those are particular cases of the general solution. – b.gates.you.know.what Sep 5 '15 at 22:53
• Those solutions are called singular solutions and DSolve does not find these. You can usually find them for first order ode's by letting the arbitrary constant tend to plus/minus infinity. – Chip Hurst Sep 6 '15 at 1:46
• Chip, I don't think my two extra solutions are "singular" in the given sense; the logistic equation satisfies the assumptions of Picard-Lindelöf everywhere, so we have existence and uniqueness of solutions everywhere. – Axel Boldt Sep 6 '15 at 2:58
• b.gatessucks: I would have to take C[1]->-Infinity to get the solution y[x]->1, but when dealing with the complex plane -Infinity and Infinity are the same thing. – Axel Boldt Sep 6 '15 at 3:02
• @Axel You are right, thanks for pointing that out. – b.gates.you.know.what Sep 6 '15 at 7:48