Differential Equations
Formation of Differential Equations
Grade 12
Question:
<p>The differential equation of the family of non-vertical lines \(y = mx + c\), where \(m\) and \(c\) are two parameters, is:</p>
<p>\(\dfrac{d^2y}{dx^2} = 0\)</p>
<p>\(\dfrac{dy}{dx} = 0\)</p>
<p>\(\dfrac{d^2y}{dx^2} + y = 0\)</p>
<p>\(\dfrac{dy}{dx} + y = 0\)</p>
Step-by-Step Solution
Key Concept: A family with two arbitrary parameters requires two eliminations using differentiation. Differentiate once to eliminate one parameter, then differentiate again to eliminate the second parameter.
<p><strong>Step 1:</strong> Start with the family equation: y = mx + c (two parameters: m and c)</p><p><strong>Step 2:</strong> Differentiate once with respect to x: dy/dx = m</p><p><strong>Step 3:</strong> Differentiate again with respect to x: d²y/dx² = 0</p><p><strong>Step 4:</strong> The equation d²y/dx² = 0 contains no arbitrary parameters and represents the differential equation of the entire family.</p><p><strong>Verification:</strong> This second-order equation is satisfied by y = mx + c for all values of m and c, confirming it's the correct differential equation.</p><p>∴ Answer: d²y/dx² = 0</p>
Correct Answer: A