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.

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