<p>Solve \(y = x\frac{dy}{dx} + \left(\frac{dy}{dx}\right)^2\).</p>
Step-by-Step Solution
Key Concept: Recognize this as Clairaut's equation in the form y = xp + f(p) where p = dy/dx. The general solution is the family of straight lines obtained by replacing p with an arbitrary constant, and a singular solution exists by eliminating p between y = xp + p² and the condition ∂/∂p(xp + p²) = 0.
<p><strong>Step 1:</strong> Recognize the equation y = x(dy/dx) + (dy/dx)² as Clairaut's equation with p = dy/dx.</p><p><strong>Step 2:</strong> <strong>General Solution:</strong> For Clairaut's equation y = xp + f(p), the general solution is y = Cx + f(C). Here, y = Cx + C² where C is an arbitrary constant.</p><p><strong>Step 3:</strong> <strong>Singular Solution:</strong> To find the singular solution, use the parametric form. Differentiate the original equation with respect to p:</p><p>∂/∂p[xp + p²] = x + 2p = 0 ⟹ x = -2p</p><p><strong>Step 4:</strong> Substitute x = -2p back into y = xp + p²:</p><p>y = (-2p)p + p² = -2p² + p² = -p²</p><p><strong>Step 5:</strong> Eliminate p: From x = -2p, we get p = -x/2. Substituting into y = -p²:</p><p>y = -(−x/2)² = -x²/4</p><p>∴ <strong>Answer:</strong> General solution: <strong>y = Cx + C²</strong> and Singular solution: <strong>y = -x²/4</strong></p>
Correct Answer: A