Question

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

• A. $-\frac{1}{y}$
• B. $\frac{1}{y}$
• C. $\frac{1}{x}$
• D. $-\frac{1}{x}$

Answer 1

The correct option is B $\frac{1}{y}$

$(x+2y^3)\frac{dy}{dx}=y$, this equation involves the term $y^3$ is not linear, if we take y as the dependent, so we can write the equation as $y\frac{dx}{dy}=x+2y^3$
$\Rightarrow$ $y\frac{dx}{dy}-x=2y^3$, which is linear differential equation in which x is as the dependent variable
$\frac{dx}{dy}-\frac{x}{y}=2y^2$
$\therefore$ I.F. $=e^{\int -\frac{1}{y}dy}=e^{-\log y}=e^{\log \frac{1}{y}}=\frac{1}{y}$