Differential Equations
Exponential growth and decay
Grade None
Question:
<p>If \(f(x) = f'(x)\) and \(f(1) = 2\), then \(\dfrac{[f(3)]}{2}\). Where [.] is G.I.F.</p>
Step-by-Step Solution
Key Concept: The condition f(x) = f'(x) is a separable differential equation that yields f(x) = Ce^x. Use the initial condition f(1) = 2 to find C, then evaluate f(3) and apply the greatest integer function.
<p><strong>Step 1:</strong> Solve the differential equation f(x) = f'(x)</p><p>Rewrite as: df/dx = f</p><p>This is separable: df/f = dx</p><p>Integrating both sides: ln|f| = x + C₁</p><p>Therefore: f(x) = Ce^x (where C = e^(C₁))</p><p><strong>Step 2:</strong> Use the initial condition f(1) = 2</p><p>f(1) = Ce¹ = 2</p><p>Ce = 2</p><p>C = 2/e</p><p><strong>Step 3:</strong> Find f(3)</p><p>f(3) = Ce³ = (2/e) · e³ = 2e²</p><p>f(3) = 2e² ≈ 2(7.389) ≈ 14.778</p><p><strong>Step 4:</strong> Apply the Greatest Integer Function (G.I.F.)</p><p>[f(3)] = [2e²] = [14.778] = 14</p><p><strong>Step 5:</strong> Calculate final answer</p><p>[f(3)]/2 = 14/2 = 7</p><p>∴ Answer: <strong>7</strong></p>
Correct Answer: 7