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Let f be a differentiable function satisfying x² * f(x) - x = 4 * integral from 0 to x of [t * f(t)] dt, with f(1) = 2/3. Find the value of 9 * f(3).
- 80
- 60
- 54
- 90
Correct answer: 80
Solution
Differentiating both sides: 2x*f(x) + x²*f'(x) - 1 = 4x*f(x). This gives f'(x) - (2/x)*f(x) = 1/x², a linear ODE. Integrating factor = x^(-2). Solution: f(x) = x² - 1/(3x). Using f(1) = 1 - 1/3 = 2/3 (confirmed). Then f(3) = 9 - 1/9 = 80/9, so 9*f(3) = 80.
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