Question

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

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

Answer 1

The correct option is C $y=\pm \sqrt{\left(lo{g}_e{x}^2\right)+2C}$

comparing the given series with serries representation of $e^x$,we get
$x=e^{xy\frac{dy}{dx}}$
$\Rightarrow logx=xy\frac{dy}{dx}$
$\Rightarrow ydy=\frac{logx}{x}dx$
$\Rightarrow ydy=logxd\left(logx\right)$
On integrating both sides, we get
$\frac{y^2}{2}=\frac{(lo{g}_{e}x{)}^2}{2}+C$
$\Rightarrow y^2=(lo{g}_{e}x{)}^2+2C$
$\Rightarrow y=\pm \sqrt{(lo{g}_{e}x{)}^2+2C}$