<p>If \(y = (x + \sqrt{1+x^2})^n\), then \((1+x^2)\dfrac{d^2y}{dx^2} + x\dfrac{dy}{dx}\) is</p>
Step-by-Step Solution
Key Concept: Recognize that y = (x + √(1+x²))^n satisfies a special differential equation. The key is to use the chain rule and observe that the derivative of (x + √(1+x²)) produces a factor of 1/√(1+x²), leading to a second-order linear ODE.
<p><strong>Step 1:</strong> Given y = (x + √(1+x²))^n. Find dy/dx using chain rule:</p><p>dy/dx = n(x + √(1+x²))^(n-1) · (1 + x/√(1+x²))</p><p>dy/dx = n(x + √(1+x²))^(n-1) · √(1+x²)/√(1+x²) · (1 + x/√(1+x²))</p><p>dy/dx = n·y/√(1+x²) · √(1+x²)/√(1+x²) = ny/√(1+x²)</p><p><strong>Step 2:</strong> Rewrite: √(1+x²)·(dy/dx) = ny</p><p><strong>Step 3:</strong> Differentiate both sides with respect to x:</p><p>√(1+x²)·(d²y/dx²) + x/√(1+x²)·(dy/dx) = n·(dy/dx)</p><p><strong>Step 4:</strong> Multiply through by √(1+x²):</p><p>(1+x²)·(d²y/dx²) + x·(dy/dx) = n√(1+x²)·(dy/dx)</p><p><strong>Step 5:</strong> From Step 2, n√(1+x²) = (1+x²)·(dy/dx)/y, so:</p><p>(1+x²)·(d²y/dx²) + x·(dy/dx) = (1+x²)·(dy/dx)²/y</p><p>∴ Answer: <strong>A</strong> (which equals n²y)</p>
Correct Answer: A