Question

The differential equation $x\, dx + y \,dy = x \,dy - y\, dx$ has solution

• A. $ \log (x^2 + y^2) = 2 \tan^{-1}\frac{y}{x}+C $
• B. $ \log (x^2 + y^2) = \tan^{-1}\frac{y}{x}+C $
• C. $ \log (x^2 + y^2) + \tan^{-1}\frac{y}{x}+C $
• D. none of these

Answer 1

Answer: (A)

$x\,dx + y\,dy = \frac{1}{2}d\left(x^{2}+y^{2}\right)$
$x\,dy-ydx = x^{2}d\left(\frac{y}{x}\right)$
$\therefore $ given differential equation can be written as
$\frac{d\left(x^{2}+y^{2}\right)}{x^{2}+y^{2}} = \frac{2x^{2}d\left(\frac{y}{x}\right)}{x^{2}+y^{2}}$
$= \frac{2d\left(\frac{y}{x}\right)}{1+\left(\frac{y}{x}\right)^{2}}$
$\Rightarrow log \left(x^{2} + y^{2}\right)$
$= 2\,tan^{-1} \frac{y}{x} +C$