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JEE MainMathematicsDefinite Integration

Let f: R R be a continuous function satisfying f(x) = x + ₀^x e^ x-t f(t) dt for all x R . Which of the following statements is true?

Options

  1. Af'(x) - f(x) = 1
  2. Bf''(x) - 2f'(x) + 1 = 0
  3. Cf''(x) + 2f'(x) - 1 = 0
  4. Df( 2) = 2 + 3 4

Correct answer

B. f''(x) - 2f'(x) + 1 = 0

Step-by-step solution

Given the integral equation: f(x) = x + ₀^x e^ x-t f(t) dt Rewrite the integral by taking e^x outside, as it is independent of the integration variable t : f(x) = x + e^x ₀^x e^ -t f(t) dt Multiply by e^ -x to isolate the integral: e^ -x f(x) - x e^ -x = ₀^x e^ -t f(t) dt Differentiating both sides with respect to x using the product rule and the Newton-Leibniz formula: -e^ -x f(x) + e^ -x f'(x) - (e^ -x - x e^ -x ) = e^ -x f(x) Divide the entire equation by e^ -x (which is always non-zero): -f(x) + f'(x) - 1 + x =

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