Question

The solution of the differential equation $(1+x^2)\frac{dy}{dx}=2x\cot y$, is:

• A. $\sec y=c(1+x^2)$
• B. $\cos y=c(1+x^2)$
• C. $\tan y=c(1+x^2)$
• D. $\cot y=c(1+x^2)$

Answer 1

Answer: (A)

$\sec y=c(1+x^2)$

Given:

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

Rearranging this equation
$\frac{dy}{\cot y}=\frac{2x}{1+x^2}dx$

Integrating both sides, we get

$\int \tan ydy=\int \frac{2x}{1+x^2}dx+c$
$\Rightarrow \log \left|\sec y\right|=\log (1+x^2)+\log c$
$\Rightarrow \log \sec y=\log c(1+x^2)$
$\Rightarrow \sec y=c(1+x^2)$

where, $c$ is the constant of integration.

Hence, option A.