Differential Equations
Formation of Differential Equations
Grade 12
Question:
<p>The differential equation of the family of curves \(y^2 = 2c(x + \sqrt{c})\), where \(c > 0\), is a parameter, is of order and degree as follows:</p>
<p>(1) order 1, degree 1</p>
<p>(2) order 1, degree 2</p>
<p>(3) order 1, degree 3</p>
<p>(4) order 2, degree 2</p>
Step-by-Step Solution
Key Concept: Eliminate the parameter c from the given family equation by differentiating it, then determine the highest derivative order and the power of that derivative in the resulting DE.
<p><strong>Step 1:</strong> Given family: y² = 2c(x + √c)</p><p>Differentiate with respect to x:</p><p>2y(dy/dx) = 2c(1 + 0) = 2c</p><p>Therefore: y(dy/dx) = c ... (1)</p><p><strong>Step 2:</strong> From equation (1), c = y(dy/dx)</p><p>Substitute back into original equation:</p><p>y² = 2y(dy/dx)[x + √(y(dy/dx))]</p><p><strong>Step 3:</strong> Simplify by dividing by 2y (y ≠ 0):</p><p>y/2 = (dy/dx)[x + √(y(dy/dx))]</p><p><strong>Step 4:</strong> Rearrange to isolate and square to eliminate the square root:</p><p>y/2 - x(dy/dx) = (dy/dx)√(y·dy/dx)</p><p>Square both sides:</p><p>[y/2 - x(dy/dx)]² = (dy/dx)² · y(dy/dx)</p><p><strong>Step 5:</strong> The resulting differential equation contains:</p><p>• Highest order derivative: dy/dx (Order = 1)</p><p>• Power of the highest order derivative: 3 (Degree = 3)</p><p>∴ Answer: Order 1, Degree 3 (Option C)</p>
Correct Answer: C