Question

Solution of $x=1+xy\left(\frac{dy}{dx}\right)+\frac{x^2{y}^2}{2!}{\left(\frac{dy}{dx}\right)}^2+$..... is

• A. $y=\log x+c$
• B. $y^2=(\log x{)}^2+c$
• C. $y=\log \left(x\right)+xy$
• D. $xy=x^y+c$

Answer 1

The correct option is B $y^2=(\log x{)}^2+c$

Given expression is in the form of $e^y$ expansion

$\Rightarrow x=e^{xy\frac{dy}{dx}}$

$\Rightarrow \log x=xy\frac{dy}{dx}$

$\Rightarrow \frac{\log x}{x}dx=ydy$

Integrate on both sides

$\Rightarrow \frac{(\log x{)}^2}{2}=\frac{y^2}{2}+c$

$\Rightarrow y^2=(\log x{)}^2+C$