Differential Equations
Linear differential equations
Grade 12
Question:
<p>If \(\dfrac{dy}{dx} + y = xe^{-x}\) and \(C = 0\), then \(f(1)\) equals:</p>
<p>\(\dfrac{1}{e}\)</p>
<p>\(\dfrac{1}{2e}\)</p>
<p>\(\dfrac{1}{2e^2}\)</p>
<p>\(\dfrac{2}{e}\)</p>
Step-by-Step Solution
Key Concept: This is a first-order linear differential equation of the form dy/dx + P(x)y = Q(x). Use integrating factor method: multiply by e^∫P(x)dx = e^x to convert the left side into a perfect derivative d/dx[e^x·y].
<p><strong>Step 1:</strong> Identify the standard form dy/dx + y = xe^(-x), where P(x) = 1 and Q(x) = xe^(-x).</p><p><strong>Step 2:</strong> Find integrating factor: IF = e^(∫1·dx) = e^x</p><p><strong>Step 3:</strong> Multiply both sides by e^x:</p><p>e^x·dy/dx + e^x·y = xe^(-x)·e^x</p><p>d/dx[e^x·y] = x</p><p><strong>Step 4:</strong> Integrate both sides:</p><p>e^x·y = ∫x dx = x²/2 + C</p><p><strong>Step 5:</strong> Apply condition C = 0 (the constant of integration):</p><p>e^x·y = x²/2</p><p><strong>Step 6:</strong> Solve for f(x):</p><p>y = (x²/2)·e^(-x)</p><p><strong>Step 7:</strong> Find f(1):</p><p>f(1) = (1²/2)·e^(-1) = (1/2)·(1/e) = 1/(2e)</p><p>∴ Answer: A</p>
Correct Answer: A