Differential Equations
General and Particular Solutions
Grade 12
Question:
<p>The solution of the equation \(\dfrac{d^2y}{dx^2} = e^{-2x}\) is</p>
<p>\(\dfrac{e^{-2x}}{4}\)</p>
<p>\(\dfrac{e^{-2x}}{4} + cx + d\)</p>
<p>\(\dfrac{1}{4}e^{-2x} + cx^2 + d\)</p>
<p>\(\dfrac{1}{4}e^{-2x} + c + d\)</p>
Step-by-Step Solution
Key Concept: Integrate the second-order differential equation twice successively, applying the constant of integration after each step to capture both arbitrary constants in the general solution.
<p><strong>Step 1:</strong> Integrate once with respect to x:</p><p>∫(d²y/dx²)dx = ∫e^(-2x)dx</p><p>dy/dx = -½e^(-2x) + C₁</p><p><strong>Step 2:</strong> Integrate again with respect to x:</p><p>∫(dy/dx)dx = ∫[-½e^(-2x) + C₁]dx</p><p>y = -½ · (-½)e^(-2x) + C₁x + C₂</p><p>y = ¼e^(-2x) + C₁x + C₂</p><p><strong>Verification:</strong> dy/dx = -½e^(-2x) + C₁ and d²y/dx² = e^(-2x) ✓</p><p>∴ Answer: <strong>y = ¼e^(-2x) + C₁x + C₂</strong> (Option B)</p>
Correct Answer: B