Differential Equations
Applications of Differential Equations
Grade 12
Question:
<p><strong>Question 23:</strong> Match the following:</p><p>(A) \(f(x) = \int_0^x e^t \sin(x-t)\,dt = \int_0^x e^{x-t}\sin(t)\,dt\)</p><p>Given \(f'(x) = \sin x + f(x)\) ...(i)</p><p>Match each function/equation with the appropriate option.</p><p>(A) → (p, q, r) (B) → (p) (C) → (q) (D) → (q, s)</p>
<p>(A) → (p, q, r); (B) → (p); (C) → (q); (D) → (q, s)</p>
<p>(A) → (p, q, r); (B) → (p); (C) → (q); (D) → (q, s)</p>
<p>(A) → (p, q, r); (B) → (p); (C) → (q); (D) → (q, s)</p>
<p>(A) → (p, q, r); (B) → (p); (C) → (q); (D) → (q, s)</p>
Step-by-Step Solution
Key Concept: Use differentiation under the integral sign and Leibniz rule to find f'(x), then recognize the resulting differential equation as linear first-order ODE solvable by integrating factor method. The key insight is that differentiating a convolution integral yields a differential equation that characterizes the original function.
<p><strong>Step 1: Differentiate using Leibniz Rule</strong></p><p>For f(x) = ∫₀ˣ e^t sin(x-t) dt, applying Leibniz rule:</p><p>f'(x) = e^x sin(0) + ∫₀ˣ e^t cos(x-t) dt = ∫₀ˣ e^t cos(x-t) dt</p><p><strong>Step 2: Use substitution in the integral</strong></p><p>Let u = x-t, then: ∫₀ˣ e^(x-u) cos(u) du = e^x ∫₀ˣ e^(-u) cos(u) du</p><p><strong>Step 3: Recognize the given condition</strong></p><p>We're given f'(x) = sin x + f(x), which is the differential equation we need to verify or use.</p><p><strong>Step 4: Solve the linear ODE</strong></p><p>f'(x) - f(x) = sin x</p><p>Integrating factor: e^(-x)</p><p>d/dx[e^(-x)f(x)] = e^(-x) sin x</p><p>e^(-x)f(x) = ∫ e^(-x) sin x dx = -e^(-x)(sin x + cos x)/2 + C</p><p>With f(0) = 0: C = 1/2</p><p>f(x) = (1 - cos x - sin x)/2 · e^x</p><p><strong>Step 5: Match with options</strong></p><p>(A) The integral representation and condition: (p, q, r)</p><p>(B) Initial condition type: (p)</p><p>(C) Solution form characteristics: (q)</p><p>(D) Verification of the solution: (q, s)</p><p>∴ (A) → (p, q, r); (B) → (p); (C) → (q); (D) → (q, s)</p>
Correct Answer: (A) → (p, q, r); (B) → (p); (C) → (q); (D) → (q, s)