For any real numbers α and β , let y α , β x , x ∈ R , be the solution of the differential equation d y d x + α y = x e 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 ?

Step-by-step solution

( d y d x +a y=x e^ x ) Integrating factor (I.F.) (= e ^ adx = e ^ ax ) So, the solution is (y e^ a x = x e^ x e^ a x d x ) ( y e^ a x = x e^ ( + ) x d x ) If ( + 0 ) ( y e^ a x =x e^ ( + ) x ( + ) - e^ ( + ) x ( + )^2 +C ) ( y= x e^ x ( + ) - e^ x ( + )^2 +C e^ -a x ) ( y= e^ x ( + ) (x- 1 a+ )+C e^ - x ) Put ( = =1 ) in (1) (y= e^x 2 (x- 1 2 )+C e^ -x ) (y(1)=1 ) (1= e 2 1 2 + C e c=e- e^2 4 ) So, (y= e^x 2 (x- 1 2 )+ (e- e^2 4 ) e^ -x ) If ( + =0 & =1 ) ( aligned & d y d x +y=x e^ -x & I.F. =e^x aligned ) Soluti