Question

The solution of $x=1+\left(xy\frac{dy}{dx}\right)+\frac{x^2{y}^2}{2!}{\left(\frac{dy}{dx}\right)}^2+\frac{x^3{y}^3}{3!}{\left(\frac{dy}{dx}\right)}^3\dots \dots$ is

• A. $y=lnx+c$
• B. $y=(lnx{)}^2+c$
• C. $y=\pm \sqrt{(lnx{)}^2+c}$
• D. None of these

Answer 1

The correct option is C $y=\pm \sqrt{(lnx{)}^2+c}$

$x=e^{xy\frac{dy}{dx}}$
$\Rightarrow lo{g}_e^x=xy\frac{dy}{dx}\Rightarrow \frac{logx}{x}dx=ydy$
$\Rightarrow \int ydy=\int \frac{lnx}{x}dx$
$\Rightarrow \frac{y^2}{2}=\frac{(logx{)}^2}{2}+C_1$
$\Rightarrow y=\pm \sqrt{(lnx{)}^2+c}$