Differential Equations
Formation of Differential Equations
Grade 12
Question:
<p>If \(y = (x + \sqrt{1 + x^2})^n\), then the differential equation satisfied by \(y\) is:</p>
<p>(1) \((1+x^2)\dfrac{d^2y}{dx^2} + x\dfrac{dy}{dx} = n^2 y\)</p>
<p>(2) \((1+x^2)\dfrac{d^2y}{dx^2} - x\dfrac{dy}{dx} = n^2 y\)</p>
<p>(3) \((1+x^2)\dfrac{d^2y}{dx^2} + x\dfrac{dy}{dx} + n^2 y = 0\)</p>
<p>(4) \((1-x^2)\dfrac{d^2y}{dx^2} + x\dfrac{dy}{dx} = n^2 y\)</p>
Step-by-Step Solution
Key Concept: Differentiate using chain rule, then observe that dy/dx contains a factor of y itself. This allows you to eliminate the expression (x + √(1+x²)) and derive a relation involving only y and its derivatives.
<p><strong>Step 1:</strong> Let y = (x + √(1+x²))ⁿ. Differentiate both sides using chain rule:</p><p>dy/dx = n(x + √(1+x²))ⁿ⁻¹ · d/dx[x + √(1+x²)]</p><p><strong>Step 2:</strong> Calculate d/dx[x + √(1+x²)] = 1 + x/√(1+x²) = (√(1+x²) + x)/√(1+x²)</p><p><strong>Step 3:</strong> Substitute back:</p><p>dy/dx = n(x + √(1+x²))ⁿ⁻¹ · (x + √(1+x²))/√(1+x²)</p><p>dy/dx = ny/(√(1+x²))</p><p><strong>Step 4:</strong> Rearrange to eliminate the radical:</p><p>√(1+x²) · dy/dx = ny</p><p>Squaring both sides: (1+x²)(dy/dx)² = n²y²</p><p><strong>Step 5:</strong> Differentiate (1+x²)(dy/dx)² = n²y² implicitly:</p><p>2x(dy/dx)² + (1+x²)·2(dy/dx)(d²y/dx²) = 2n²y(dy/dx)</p><p>Dividing by 2(dy/dx):</p><p><strong>(1+x²)d²y/dx² + x·dy/dx - n²y = 0</strong></p><p>∴ Answer: A</p>
Correct Answer: A