Solve Homogeneous DE Easier: Better Substitution?

[Werg22]

Given y' = y / (x + y^2), the substitution u = y^2 will give a homogeneous DE which can then be easily solved. Is there a substitution which would make things easier?

 

[rock.freak667]

Try V=y/x

But it is kinda long in my opinion.


EDIT: The easiest way is your substitution of u=y^{-2}, anything else, is just harder.

 

[Werg22]

I think the substitution u = y^2 + x is better. I haven't tried it though.

 

[Matthew Rodman]

There is a solution that does not involve a substitution... if that's any help...

First, multiply through by x + y^2, to get

x y^{\prime} + y^2 y^{\prime} = y

rearrange to get

x y^{\prime} - y = -y^2 y^{\prime}

but

x y^{\prime} - y = y^2 ( \phi - \frac{x}{y})^{\prime}

(where \phi is a constant.) So,

( \phi - \frac{x}{y})^{\prime} = -y^{\prime}

which you can integrate to get

\phi - \frac{x}{y} = - y

which you can turn into a quadratic by multiplying through by y, leaving you with.

y(x) = \frac{-\phi \pm \sqrt{\phi^2 + 4x}}{2}