# Hi! how to solve? y(y-2x(y'))^3=(y')^2 (1 Viewer)

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Hello Everybody! Have any suggestions how to solve this differential equation, I haven't.. :(
y(y-2x(y'))^3=(y')^2

#### HallsofIvy

The only thing I can suggest is that you first treat it as a cubic equation for y' and try to solve for y'. But that obviously is going to be difficult!

I tried this way, but it didn't give me any positive result. Moreover even Maple can't solve it exact.. it gives me huge expression..

#### John Creighto

I tried this way, but it didn't give me any positive result. Moreover even Maple can't solve it exact.. it gives me huge expression..
Are numeric or series solutions okay?

The right solutions are y^2=2*(C^3)*x + C^2 and 27*(x^2)*(y^2)=1. Hmm.. Lagrange method doesn't fit properly here

#### coomast

Hello Everybody! Have any suggestions how to solve this differential equation, I haven't.. :(
y(y-2x(y'))^3=(y')^2

This equation can be solved by the following method. If a DE of the first order but not first degree can be written as $x-\phi(y,y')=0$ the way to solve it is by taking the derivative of it and consider y and y' as variables. Then you need to substitute dy/y' for dx and you get a first degree DE in y and y'. After integrating this equation you need to eliminate y' between this solution and the original equation to obtain the general solution. In this process you need to make sure that any factors that are eliminated are to be looked upon with care afterwards because they can be singular solutions. Let's try this on your problem.

OK, first rewriting the original DE gives:

$$x=\frac{y}{2y'}-\frac{1}{2y^{1/3}y'^{1/3}}$$

The derivative of this is:

$$dx=\frac{1}{2}\left[\frac{1}{y'}+\frac{1}{3y'^{1/3}y^{4/3}}\right]dy+ \frac{1}{2}\left[-\frac{y}{y'^2}+\frac{1}{3y^{1/3}y'^{4/3}}\right]dy'$$

Substituting $\frac{dy}{y'}=dx$ gives after rewriting:

$$\frac{1}{2y'}\left[1-\frac{y'^{2/3}}{3y^{4/3}}\right]dy= -\frac{y}{2y'^2}\left[1-\frac{y'^{2/3}}{3y^{4/3}}\right]dy'$$

The factor between brackets can be eliminated but is important for later on to see if it is a singular solution. We have thus so far:

$$\frac{dy'}{y'}=-\frac{dy}{y}$$

Which has as solution:

$$y'=\frac{C}{y}$$

Substituting this in the original equation gives you:

$$y\left(y-2x\frac{C}{y}\right)^3=\frac{C^2}{y^2}$$

Or, when we set C=K^3:

$$y^2=K^2+2xK^3$$

This is the general solution to the DE. Now the factor that was eliminated needs to be looked upon. Setting it equal to zero:

$$1-\frac{y'^{2/3}}{3y^{4/3}}=0$$

Or, after rewriting:

$$y'=3\sqrt{3}y^2$$

Which has the solution:

$$-\frac{1}{y}=3\sqrt{3}x$$

Or:

$$27x^2y^2=1$$

And it is not possible to get this by setting K to any number and thus is a singular solution because it is also a solution to the original DE.

Hope this helps, coomast

Thanks, Coomast, for help.

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