સાબિત કરો કે વિકલ સમીકરણ એક સમપરિમાણીય સમીકરણ છે અને તેનો ઉકેલ શોધો:
$\left\{x \cos \left(\frac{y}{x}\right)+y \sin \left(\frac{y}{x}\right)\right\} y \, dx = \left\{y \sin \left(\frac{y}{x}\right)-x \cos \left(\frac{y}{x}\right)\right\} x \, dy$

આપેલ વિકલ સમીકરણ:
$\left\{x \cos \left(\frac{y}{x}\right)+y \sin \left(\frac{y}{x}\right)\right\} y \, dx = \left\{y \sin \left(\frac{y}{x}\right)-x \cos \left(\frac{y}{x}\right)\right\} x \, dy$
$\frac{dy}{dx} = \frac{\left\{x \cos \left(\frac{y}{x}\right)+y \sin \left(\frac{y}{x}\right)\right\} y}{\left\{y \sin \left(\frac{y}{x}\right)-x \cos \left(\frac{y}{x}\right)\right\} x} \quad \dots(1)$
ધારો કે $F(x, y) = \frac{\left\{x \cos \left(\frac{y}{x}\right)+y \sin \left(\frac{y}{x}\right)\right\} y}{\left\{y \sin \left(\frac{y}{x}\right)-x \cos \left(\frac{y}{x}\right)\right\} x}$
$F(\lambda x, \lambda y) = \frac{\left\{\lambda x \cos \left(\frac{\lambda y}{\lambda x}\right)+\lambda y \sin \left(\frac{\lambda y}{\lambda x}\right)\right\} \lambda y}{\left\{\lambda y \sin \left(\frac{\lambda y}{\lambda x}\right)-\lambda x \cos \left(\frac{\lambda y}{\lambda x}\right)\right\} \lambda x} = \lambda^0 F(x, y)$
આમ,સમીકરણ સમપરિમાણીય છે.
$y = vx$ લેતા,$\frac{dy}{dx} = v + x \frac{dv}{dx}$.
$v + x \frac{dv}{dx} = \frac{v \cos v + v^2 \sin v}{v \sin v - \cos v}$
$x \frac{dv}{dx} = \frac{2v \cos v}{v \sin v - \cos v}$
$\left(\tan v - \frac{1}{v}\right) dv = \frac{2}{x} dx$
બંને બાજુ સંકલન કરતા:
$\log(\sec v) - \log v = 2 \log x + \log C$
$\log\left(\frac{\sec v}{v}\right) = \log(Cx^2)$
$\sec \left(\frac{y}{x}\right) = Cxy$
$\cos \left(\frac{y}{x}\right) = \frac{1}{Cxy} \implies xy \cos \left(\frac{y}{x}\right) = k$.