Question

The integrating factor of the differential equation $\left(1+x^{2}\right)\frac{dy}{dx}+xy=cos\,x$ is equal to

• A. $\sqrt{1+x}$
• B. $\sqrt{1+2x^{2}}$
• C. $\sqrt{1+x^{2}}$
• D. $\sqrt{2+x^{2}}$

Answer 1

Answer: (C)

$\sqrt{1+x^{2}}$

Answer 2

We have, $(1 + x^2) \frac{dy}{dx} + xy = cos\,x$
$\Rightarrow \frac{dy}{dx} + (\frac{x}{1+x^2}) y =\frac{cos\,x}{1+x^2}$
$\therefore I.F. = e^{\left(\frac{1}{2}\int\frac{2x}{1+x^{2}}dx\right)} = e^{\frac{1}{2}log\left(1+x^{2}\right)}$
$= \sqrt{1+x^{2}}$