Question

Integrating Factor (IF) of the differential equation $\frac{dy}{dx}-\frac{3x^2}{1+x^3}\left(y\right)=\frac{{\sin }^2\left(x\right)}{1+x}$ is

• A. $e^{1+x^3}$
• B. $\log (1+x^3)$
• C. $1+x^3$
• D. $\frac{1}{1+x^3}$

Answer 1

Answer: (D)

$\frac{1}{1+x^3}$

Given equation is
$\frac{dy}{dx}-\frac{3x^2}{1+x^3}\left(y\right)=\frac{{\sin }^2\left(x\right)}{1+x}$

$\therefore IF=e^{\int Pdx}=e^{-\int \frac{3x^2}{1+x^3}dx}$

$\therefore IF=e^{\int Pdx}=e^{-\int \frac{3x^2}{1+x^3}dx}$

$=e^{-\log (1+x^3)}={(1+x^3)}^{-1}$

$=\frac{1}{1+x^3}$