For any real numbers \alpha and \beta, let y_{\alpha, \beta}(x), x \in \mathbb{R}, be the solution of the differential equation \frac{d y}{d x}+\alpha y=x e^{\beta x}, y(1)=1 Let S=\{y_{\alpha, \beta}(x): \alpha, \beta \in\mathbb{R}\}. Then which of the following functions belong(s) to the set S ? [JEE Adv 2021 P2]

f(x)=\frac{x^{2}}{2} e^{-x}+(e-\frac{1}{2}) e^{-x}

f(x)=-\frac{x^{2}}{2} e^{-x}+(e+\frac{1}{2}) e^{-x}

f(x)=\frac{e^{x}}{2}(x-\frac{1}{2})+(e-\frac{e^{2}}{2}) e^{-x}

f(x)=\frac{e^{x}}{2}(\frac{1}{2}-x)+(e+\frac{e^{2}}{2}) e^{-x}

JEE Advanced 2021 Paper 2

Section: Mathematics

Question : 42 of 57

Marks: +1, -0

For any real numbers α and β, let yα,β(x),x∈ℝ, be the solution of the differential equation ‌

+αy=xeβx,y(1)=1 Let S={yα,β(x):α,β∈ℝ}. Then which of the following functions belong(s) to the set S ?

[JEE Adv 2021 P2]

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