Question

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

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

Answer 1

The correct option is C $y=\pm \sqrt{({log}_{e}x{)}^2+2C}$

We have $x=e^{xy.dy/dx}$
$\Rightarrow \log x=xy\frac{dy}{dx}\Rightarrow ydy=\frac{\log x}{x}dx$
On integration, we get
$\frac{y^2}{2}=\frac{({\log }_{e}x{)}^2}{2}+C$
$\Rightarrow y^2=({\log }_{e}x{)}^2+2C$
Hence $y=\pm \sqrt{({\log }_{e}x{)}^2+2C}$