Solve the differential equation: y′′y′+yy′+yy′′=0

[Baal Hadad]

Homework Statement

Solve the following equation:

Relevant Equations

$y''y'+yy'+yy''=0$

I tried the substitution $y=e^{\int z(x)}$,$z(x)$ is an arbitrary function to be determined.

Substitute this to the original differential equation,and dividing $y^2$ yields $(z+1)z'+z^3+z^2+z=0$,which is a first order differential equation.

Trying to solve this first order differential equation yields $x= ln z -\frac { ln (z^2+z+1)}{2} +\frac {\arctan( {\frac {2z+1}{\sqrt{3}}})}{{\sqrt{3}}}$

Then I can't continue here.The expression seems implicit and I cannot get an expression of $z$ in terms of $x$,that can be substituted back to $y=e^{\int z(x)}$.Had I use the wrong substitution or any better subsitutions?

Thanks.

 

[fresh_42]

Are you sure you haven't forgotten to mention something? WolframAlpha gives a monster as solution, so I checked some simple solutions. Of course we have all constants $y(x)\equiv c$, and $y(x)=ae^{bx}$ yields $b=-\frac{1}{2}\pm i\frac{\sqrt{3}}{2}$. And these are the easy ones where all three terms are more or less equal. Using the symmetries of the equation gave me something like $1=(1+u(x))(1+v(x))$ with $u(x)=\frac{y}{y''}\, , \,v(x)=\frac{y'}{y''}$. So without additional assumptions on $y''$ this seems quite complicated. This might be most obvious in the notation $u+v+uv=0$, which equals a sum and a product, two very different objects.

 

[Baal Hadad]

Are you sure you haven't forgotten to mention something?

Sorry,please tell me if there is anything that I should mention but I forgot.

So,do you mean that this equation cannot be solved analytically?

By the way,I tried this substitution from this :

 

[fresh_42]

It can with simple solutions as those I mentioned. I doubt there is an expression for all solutions.
Have a look: https://www.wolframalpha.com/input/?i=y"y'+yy'+yy"=0

 

[Baal Hadad]

It can with simple solutions as those I mentioned. I doubt there is an expression for all solutions.
Have a look: https://www.wolframalpha.com/input/?i=y"y'+yy'+yy"=0

Well,I also doubt that there exist any methods to produce such an expression as shown.
According to Wolfram Alpha ,I think actually the main part of the solution consist of the inverse function of the implicit expression above.Maybe my thought still has a little bit of use...

 

[Baal Hadad]

Well,I also doubt that there exist any methods to produce such an expression as shown.
According to Wolfram Alpha ,I think actually the main part of the solution consist of the inverse function of the implicit expression above.Maybe my thought still has a little bit of use...

Or can I integrate the inverse function,even I don't know it?