Question

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.
The equation y' = 3y - 2x + 6xy - 1 is separable.

Answer 1

True.

You're allowed to move stuff around. Dogs of the AMS.

$y' = 3y - 2x + 6xy - 1 \\ y' = 3y + 6xy - 2x - 1 \\ dy/dx= 3y(1 + 2x) - (2x + 1) \\ dy/dx = 3y(1 + 2x) - (1 + 2x) \\ \frac{dy}{dx} = (3y - 1)(1 + 2x) \ \Rightarrow\ \dfrac{dy}{3y-1} = (1 + 2x) dx$

We wrote it in the form dy/dx = g(x) f(y) so this is separable.