First Order ODE/x dy = (y + x^2 + 9 y^2) dx/Proof 1

Proof
Divide both sides of $(1)$ by $x^2 \, \mathrm d x$ to get:
 * $\dfrac 1 x \dfrac {\mathrm d y} {\mathrm d x} = \dfrac 1 x \left({\dfrac y x }\right) + 1 + 9 \left({\dfrac y x}\right)^2$

Now apply the substitution:
 * $y = u x$

This implies then that:
 * $\dfrac {\mathrm d y} {\mathrm d x} = u + x \dfrac {\mathrm d u} {\mathrm d x}$

Now substitute everything into $(1)$ to get:

Now it becomes Separation of Variables and we end up with:

Substitute back for $u$:


 * $\tan^{-1} \left({\dfrac {3 y} x}\right) = 3 x + K$