I am faced with this d.e: $$y' = \frac{2xy+3y^2}{2xy +x^2}$$
I know these methods for solving a $1^{st}$ order, o.d.e:
*Integrating factor *Separation of variables *Exact d.e *Using intgrating factor to transform a d.e to exact d.e
I could try each of these methods in a brute force way until i manage to find a solution for my d.e. But this is not optimal.
So how do I know which method to use? In this example and/or in general.
$$y' = \frac{2xy+3y^2}{2xy +x^2}$$ is a homogeneous equation.
The change of variable $$y=ux$$ will transform the equation into a separable one.
$$ u+xu'=\frac {2u+3u^2}{2u+1} $$
which simplifies to $$ \frac{dx}{x}=\frac{(2u+1)du}{u^2+u}$$