<p>The differential equation of the family of parabolas having their axis as x-axis is \(y^2 = 4a(x - h)\). The order and degree of this differential equation are:</p>
Step-by-Step Solution
Key Concept: A family with two arbitrary constants (a and h) requires two differentiations to eliminate them, determining the order. The degree is the power of the highest order derivative after eliminating constants.
<p><strong>Step 1:</strong> Given family: y² = 4a(x - h) with two arbitrary constants (a, h)</p><p><strong>Step 2:</strong> First differentiation with respect to x:<br/>2y(dy/dx) = 4a<br/>⟹ a = y(dy/dx)/2</p><p><strong>Step 3:</strong> Second differentiation with respect to x:<br/>2(dy/dx)² + 2y(d²y/dx²) = 0<br/>⟹ (dy/dx)² + y(d²y/dx²) = 0</p><p><strong>Step 4:</strong> This is a second-order DE (highest derivative is d²y/dx²)<br/>The highest order derivative has power 1, so degree = 1</p><p><strong>Step 5:</strong> Verify constants are eliminated: The equation 2y(dy/dx)² + 2y²(d²y/dx²) = 0 contains no arbitrary constants.</p><p>∴ <strong>Order = 2, Degree = 1</strong></p><p><strong>Answer: A</strong></p>
Correct Answer: A