Question

Integrating factor of the differential equation
$\left(1-x^2\right)\frac{dy}{dx}-xy=1$ is

• A. $-x$
• B. $\frac{x}{1+x^2}$
• C. $\sqrt{1-x^2}$
• D. $\frac{1}{2}log\left(1-x^2\right)$

Answer 1

Answer: (C)

$\left(1-x^2\right)\frac{dy}{dx}-xy=1$
$\Rightarrow \frac{dy}{dx}-\frac{x}{1-x^2}\cdot y=\frac{1}{1-x^2}$
$I.F.=e^{-\int \frac{x}{1-x^2}dx}=e^{\frac{1}{2}\int \frac{-2xdx}{1-x^2}}$
$=e^{\frac{1}{2}log\left(1-x^2\right)}=e^{log{\left(1-x^2\right)}^{\frac{1}{2}}}$
$=\sqrt{1-x^2}$