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A function y = f(x) satisfies (x+1) f'(x) - 2(x² + x) f(x) = e^(x²)/(x+1) for all x > -1, with f(0) = 5. Find f(x).

1. ((3x + 5)/(x + 1)) * e^(x²)
2. ((6x + 5)/(x + 1)) * e^(x²)
3. ((6x + 5)/((x + 1)²)) * e^(x²)
4. ((5 - 6x)/(x + 1)) * e^(x²)

Correct answer: ((6x + 5)/(x + 1)) * e^(x²)

Solution

Reducing to f' - 2x f = e^(x²)/(x+1)² and using IF e^(-x²) gives f*e^(-x²) = -1/(x+1) + C, fixed by f(0)=5.

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