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Application of Integration question

2021 · Shift 2 · Q23
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  5. /2021 · Shift 2 · Q23

Application of Integration question

2021 · Shift 2 · Q23

JEE AdvancedMathematicsApplication of IntegrationMultiple correct+4 / −2
For any real numbers α\alphaα and β\betaβ, let yα,β(x){y_{\alpha ,\beta }}(x)yα,β​(x), x ∈\in∈ R, be the solution of the differential equation dydx+αy=xeβx,y(1)=1{{dy} \over {dx}} + \alpha y = x{e^{\beta x}},y(1) = 1dxdy​+αy=xeβx,y(1)=1. Let S={yα,β(x):α,β∈R}S = \{ {y_{\alpha ,\beta }}(x):\alpha ,\beta \in R\}S={yα,β​(x):α,β∈R}. Then which of the following functions belong(s) to the set S?
  1. A
    f(x)=x22e−x+(e−12)e−xf(x) = {{{x^2}} \over 2}{e^{ - x}} + \left( {e - {1 \over 2}} \right){e^{ - x}}f(x)=2x2​e−x+(e−21​)e−x
  2. B
    f(x)=−x22e−x+(e+12)e−xf(x) = - {{{x^2}} \over 2}{e^{ - x}} + \left( {e + {1 \over 2}} \right){e^{ - x}}f(x)=−2x2​e−x+(e+21​)e−x
  3. C
    f(x)=ex2(x−12)+(e−e24)e−xf(x) = {{{e^x}} \over 2}\left( {x - {1 \over 2}} \right) + \left( {e - {{{e^2}} \over 4}} \right){e^{ - x}}f(x)=2ex​(x−21​)+(e−4e2​)e−x
  4. D
    f(x)=ex2(12−x)+(e+e24)e−xf(x) = {{{e^x}} \over 2}\left( {{1 \over 2} - x} \right) + \left( {e + {{{e^2}} \over 4}} \right){e^{ - x}}f(x)=2ex​(21​−x)+(e+4e2​)e−x
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