Question

The integrating factor for the differential equation
$\frac{dy}{dx}(x+2y^3)=y$ is

Answer 1

Let's take the given differential equation,
$\frac{dy}{dx}(x+2y^3)=y$
$\Rightarrow (x+2y^3)=y\frac{dx}{dy}$
$\Rightarrow \frac{x}{y}+2y^2=\frac{dx}{dy}$
$\Rightarrow \frac{dx}{dy}-\frac{x}{y}=2y^2$

Compare it with $\frac{dx}{dy}+Px=Q$ we get,
$P=-\frac{1}{y}$
$I.F.=e^{\int Pdy}$
Now,
$\int Pdy=-\frac{1}{y}dy$
$=-{\log }_{e}y={\log }_e\frac{1}{y}$
$\therefore I.F.=e^{{\log }_e\frac{1}{y}}$
$=\frac{1}{y}$