Find the necessary and sufficient condition for the…

Find the necessary and sufficient condition for the solutions of the homogeneous equation $y'=f\!\left(\dfrac{y}{x}\right)$ to be closed curves surrounding the origin.

Step 1: Convert to polar form

Differentiating $x=r\cos\theta,\ y=r\sin\theta$ and substituting into $y'=f(y/x)=f(\tan\theta)$ and solving for $r'=dr/d\theta$ gives

\[\frac{1}{r}\frac{dr}{d\theta}=\frac{\cos\theta+f(\tan\theta)\sin\theta}{f(\tan\theta)\cos\theta-\sin\theta}\;\equiv\;h(\theta).\]

Step 2: Closure condition

Integrating, $\ln r(\theta)=\displaystyle\int h(\theta)\,d\theta$. The solution curve through a given point is a closed loop encircling the origin precisely when $r$ returns to its starting value after $\theta$ increases by $2\pi$, i.e. when the net change in $\ln r$ over one revolution vanishes:

\[\boxed{\,\int_0^{2\pi}\frac{\cos\theta+f(\tan\theta)\sin\theta}{f(\tan\theta)\cos\theta-\sin\theta}\,d\theta=0.}\]