Question

Assertion :Integrating factor of $(1+x^2)\frac{dy}{dx}+xy=\frac{1}{2x}$ is given by $\sqrt{1+x^2}$ Reason: integrating factor of $\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)$ is $e^{\int P\left(x\right)dx}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Given equation is $\frac{dy}{dx}(1+x^2)+xy=\frac{1}{2x}$

$\Rightarrow$ $\frac{dy}{dx}+\frac{x}{1+x^2}y=\frac{1}{2x(1+x^2)}$

$\therefore$ I.F.$=e^{\int \frac{x}{1+x^2}dx}=e^{\int \frac{1}{2}\left(\frac{2x}{1+x^2}\right)dx}$

$=e^{\frac{1}{2}\log (1+x^2)}=e^{\log {(1+x^2)}^{\frac{1}{2}}}$

$=\sqrt{1+x^2}$
Hence both statement are correct and Assertion is followed by reason.