Question

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

• A. $y={\log }_e\left(x\right)+C$
• B. $y={\left({\log }_{e}x\right)}^2+C$
• C. $y=\pm \sqrt{{\left({\log }_{e}x\right)}^2+2C}$
• D. $xy=x^y+K$

Answer 1

Answer: (C)

$y=\pm \sqrt{{\left({\log }_{e}x\right)}^2+2C}$

Given Differential equation is the expansion of $e^{xy\frac{dy}{dx}}$

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

Taking log on both sides, We get

$logx=xy\frac{dy}{dx}$

$\frac{logx}{x}dx=ydy$

Integrating both sides

$\Rightarrow \frac{(logx{)}^2}{2}+C=\frac{y^2}{2}$

$\Rightarrow y=\pm \sqrt{(lo{g}_{e}x{)}^2+2C}$