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

Question

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

dy

dx

+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]

Answer

f(x)=‌

x2

2

e−x+(e−‌

1

2

)e−x

Answer

f(x)=−‌

x2

2

e−x+(e+‌

1

2

)e−x

Answer

f(x)=‌

ex

2

(x−‌

1

2

)+(e−‌

e2

2

)e−x

Answer

f(x)=‌

ex

2

(‌

1

2

−x)+(e+‌

e2

2

)e−x

JEE Advanced 2021 Paper 2