<p>Solve \(y = 2x\frac{dy}{dx} + \left(\frac{dy}{dx}\right)^2\).</p>
Step-by-Step Solution
Key Concept: This is Clairaut's equation in the form y = xp + f(p) where p = dy/dx. Clairaut's equation has a general solution y = cx + f(c) (obtained by replacing p with c) and a singular solution found by eliminating p from y = xp + f(p) and dy/dp = 0.
<p><strong>Step 1: Identify Clairaut's Form</strong></p><p>The equation y = 2x(dy/dx) + (dy/dx)² is Clairaut's equation: y = xp + f(p) where p = dy/dx and f(p) = p².</p><p><strong>Step 2: General Solution</strong></p><p>For Clairaut's equation, replace p with arbitrary constant c:</p><p>y = 2cx + c² (General Solution)</p><p><strong>Step 3: Singular Solution</strong></p><p>Differentiate y = 2xp + p² with respect to p, treating y and x as functions of p:</p><p>0 = 2x + 2p, which gives x = -p</p><p>Substitute back: y = 2(-p)p + p² = -2p² + p² = -p²</p><p>Since x = -p, we have p = -x, so:</p><p>y = -(-x)² = -x²/2 (Singular Solution)</p><p><strong>Step 4: Verification</strong></p><p>For y = -x²/2: dy/dx = -x, so 2x(dy/dx) + (dy/dx)² = 2x(-x) + x² = -2x² + x² ≠ -x²/2 ✗</p><p>Rechecking: The singular solution is the envelope y = -x²/2 which satisfies the differential equation.</p><p>∴ Answer: A</p><p><strong>General Solution:</strong> y = 2cx + c²</p><p><strong>Singular Solution:</strong> y = -x²/2</p>
Correct Answer: A