Question

Solution of the differential equation $xy\frac{dy}{dx}=\frac{1 + y^{2}}{1 + x^{2}}\left(1 + x + x^{2}\right)$ , given that $x=1,y=0$ is

• A. $\ell n\sqrt{1 + y^{2}}=\ell nx+tan^{- 1}x-\frac{\pi }{2}$
• B. $ln\left(\frac{1 + y^{2}}{x^{2}}\right)=2\left(tan\right)^{- 1}x-\frac{\pi }{2}$
• C. $\ell n\left(\frac{1 + y^{2}}{x^{2}}\right)=\frac{\pi }{4}-2\left(tan\right)^{- 1}x$
• D. $\ell n\left(\frac{1 + y^{2}}{x^{2}}\right)=\left(tan\right)^{- 1}x-\frac{\pi }{4}$

Answer 1

Answer: (B)

$\int \frac{ ydy }{1+ y ^{2}}=\int \frac{1+ x + x ^{2}}{ x \left(1+ x ^{2}\right)} dx$
$\frac{1}{2} \ell n \left(1+ y ^{2}\right)=\tan ^{-1} x +\ell nx + c$
$\Rightarrow \ell n \left(\frac{1+ y ^{2}}{ x ^{2}}\right)=2 \tan ^{-1} x + c$