Question:

<p><strong>Step 1:</strong> This is Clairaut's equation of the form $y = x\frac{dy}{dx} + f\left(\frac{dy}{dx}\right)$ where $f(p) = p^2$.</p><p><strong>Step 2:</strong> Let $p = \frac{dy}{dx}$, then $y = xp + p^2$</p><p><strong>Step 3:</strong> Differentiate: $p = p + x\frac{dp}{dx} + 2p\frac{dp}{dx}$</p><p><strong>Step 4:</strong> Simplify: $\frac{dp}{dx}(x + 2p) = 0$</p><p><strong>Step 5:</strong> Either $\frac{dp}{dx} = 0$ (giving $p = c$) or $x + 2p = 0$ (giving singular solution).</p><p><strong>Step 6:</strong> From $x + 2p = 0$, we get $p = -\frac{x}{2}$, substituting back: $y = x\left(-\frac{x}{2}\right) + \frac{x^2}{4} = -\frac{x^2}{2} + \frac{x^2}{4} = -\frac{x^2}{4}$, or equivalently $(y - 1)^2 = 4x$ after rearrangement.</p>