Medium Practice — Set 1

Quick Recap — Separable & Homogeneous

• Homogeneous $\tfrac{dy}{dx}=F\big(\tfrac yx\big)$: substitute $y=vx$, so $\tfrac{dy}{dx}=v+x\tfrac{dv}{dx}$.
• Formation: eliminate the arbitrary constants by differentiating as many times as there are constants.
• $\tfrac{dy}{dx}=-\tfrac xy\Rightarrow x^2+y^2=C$ (circles); $\tfrac{dy}{dx}=\tfrac yx\Rightarrow y=Cx$ (lines through origin).
• Apply the initial condition only after integrating.