Question

The solution of differential equation
$4xy\frac{dy}{dx}=\frac{3{(1+x)}^2(1+y^2)}{(1+x^2)}$ is

• A. $\log (1+y)=\log x+2\tan x+constant$
• B. $\log (1+y^2)=3\log \left(\frac{1}{x}\right)+6{\tan }^{-1}x+constant$
• C. $2\log (1+y^2)=3\log x+6{\tan }^{-1}x+constant$
• D. None of the above

Answer 1

The correct option is C $2\log (1+y^2)=3\log x+6{\tan }^{-1}x+constant$

Given, $4xy\frac{dy}{dx}=\frac{3{(1+x)}^2(1+y^2)}{(1+x^2)}$

$\Rightarrow \frac{4ydy}{1+y^2}=\frac{3{(1+x)}^2}{x\left({1+x}^2\right)}dx$

$\Rightarrow \frac{4ydy}{1+y^2}=(\frac{3}{x}+\frac{6}{{1+x}^2})dx$

Integrating both sides

$\Rightarrow \int \frac{4ydy}{1+y^2}=\int (\frac{3}{x}+\frac{6}{{1+x}^2})dx$

$\Rightarrow 2\log (1+y^2)=3\log x+6{\tan }^{-1}x+C$