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

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

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

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

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

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

Substitute back for $u$:


 * $\tan^{-1} \paren {\dfrac {3 y} x} = 3 x + C$