What is the solution to a 3rd order nonlinear ODE?

[Max0526]

Hi, everybody.
I have an ODE:
$$y'''+2y''y-3y'^2=0$$
I know that it has an analytic solution, but I cannot get it (yet).
Can anybody help me?
(I don't need a full explanation how to solve it, just some hints or just the solution with 3 arbitrary constants).
Thanks beforehand,
Max.

 

[Reb]

Hi, everybody.
I have an ODE:
$$y'''+2y''y-3y'^2=0$$
I know that it has an analytic solution, but I cannot get it (yet).
Can anybody help me?
(I don't need a full explanation how to solve it, just some hints or just the solution with 3 arbitrary constants).
Thanks beforehand,
Max.

Three constants? That would be the case if the ODE was linear.

There is another problem here. The ODE is autonomous, and all functions of the form y=const. are solutions, plus the solutions cannot meet each other. So this ODE has only trivial solutions.

 

[g_edgar]

How about:
$$y = \frac{6}{x+a}$$

 

[Reb]

How about ...you re right?
I mistakingly assumed that every solution has to meet the y axis.

 

[kosovtsov]

Hi, everybody.
I have an ODE:
$$y'''+2y''y-3y'^2=0$$
I know that it has an analytic solution, but I cannot get it (yet).
Can anybody help me?
(I don't need a full explanation how to solve it, just some hints or just the solution with 3 arbitrary constants).
Thanks beforehand,
Max.

You are right, this ODE has an analytic general solution, but it is not so simple and includes hypergeom functions and so on. To see the solution in implicit form use Maple with

ode:=diff(y(x),x,x,x)+2*diff(y(x),x,x)*y(x)-3*diff(y(x),x)^2=0;

ans:=dsolve(ode);