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)

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