<p>The differential equation which represents the family of curves \(y = c_1 e^{c_2 x}\), where \(c_1\) and \(c_2\) are arbitrary constants is</p>
Step-by-Step Solution
Key Concept: To eliminate two arbitrary constants, differentiate the curve equation twice to get three equations, then use algebraic elimination to derive a relationship independent of c₁ and c₂.
<p><strong>Step 1:</strong> Start with the family of curves: <em>y</em> = c₁e^(c₂x)</p><p><strong>Step 2:</strong> Differentiate once with respect to x:<br/><em>dy</em>/<em>dx</em> = c₁c₂e^(c₂x) = c₂y</p><p><strong>Step 3:</strong> From Step 2, observe that <em>dy</em>/<em>dx</em> = c₂y, which gives c₂ = (1/<em>y</em>)·<em>dy</em>/<em>dx</em></p><p><strong>Step 4:</strong> Differentiate Step 2 again:<br/><em>d</em>²<em>y</em>/<em>dx</em>² = c₂·<em>dy</em>/<em>dx</em></p><p><strong>Step 5:</strong> Substitute c₂ = (1/<em>y</em>)·<em>dy</em>/<em>dx</em> into Step 4:<br/><em>d</em>²<em>y</em>/<em>dx</em>² = [(1/<em>y</em>)·<em>dy</em>/<em>dx</em>]·<em>dy</em>/<em>dx</em> = (1/<em>y</em>)·(<em>dy</em>/<em>dx</em>)²</p><p><strong>Step 6:</strong> Rearrange to get:<br/><em>y</em>·<em>d</em>²<em>y</em>/<em>dx</em>² = (<em>dy</em>/<em>dx</em>)²</p><p>∴ <strong>Answer: D</strong></p>
Correct Answer: D