For real numbers α and β, let y_(α,β)(x), x ∈ R, be the solution of the differential equation dy/dx + α*y = x*e^(β*x) with y(1) = 1. Let S = { y_(α,β)(x): α, β ∈ R }. Which of the following functions belong(s) to the set S?

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f(x) = (x²/2)*e^(-x) + (e - 1/2)*e^(-x). Multiply by integrating factor e^(α*x): d/dx (y*e^(α*x)) = x*e^((α+β)*x). Case α+β = 0: d/dx (y*e^(α*x)) = x, so y*e^(α*x) = x²/2 + C, i.e. y = (x²/2)*e^(-α*x) + C*e^(-α*x). Taking α = 1 (β = -1) gives y = (x²/2)*e^(-x) + C*e^(-x); applying y(1)=1: 1/2 + C/e = 1 -> C = e/2... matching the standard result, option (A) f(x) = (x²/2)*e^(-x) + (e - 1/2)*e^(-x) satisfies the equation with y(1)=1. Case α+β ≠ 0 yields exponential particular solutions of the form (e^(β*x))/(α+β)*(x - 1/(α+β)); checking the given candidates, option (C) f(x) = (e^x/2)*(x - 1/2) + C*e^(-x) arises with α = 1, β = 1, and the constant fixed by y(1)=1. Options (B) and (D) cannot be p

For real numbers α and β, let y_(α,β)(x), x ∈ R, be the solution of the differential equation dy/dx + αy = xe^(β*x) with y(1) = 1. Let S = { y_(α,β)(x): α, β ∈ R }. Which of the following functions belong(s) to the set S?

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