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For any real numbers α and β , let y α , β (x), x ∈ R, be the solutions of the differential equation dx dy + α y = xe β x , y(1) = 1 Let S = {y α , β (x) : α , β ∈ R}. Then which of the following functions belong(s) to the set S?

1. f(x) = 2 x 2 e –x + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − 2 1 e e –x
2. f(x) = – 2 x 2 e –x + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + 2 1 e e –x
3. f(x) = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − 2 1 x 2 e x + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − 4 e e 2 e –x
4. f(x) = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − x 2 1 2 e x + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + 4 e e 2 e –x

Correct answer: f(x) = 2 x 2 e –x + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − 2 1 e e –x

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