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 ?
C. f x = e x 2 x - 1 2 + e - e 2 4 e - x
\( d y d x +a y=x e^ x \) 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 \( + =0 \& =1\) \( aligned & d y d x +y=x e^ -x \\ & I.F. =e^x aligned \) Solution is \(y e^x= x d x\) \( y e^x= x^2 2 +C\) \(y= x^2 2 e^ -x +C e^ -x \) \(y(1)=1\) \(1= 1 2 e + C e C=e- 1 2 \) \(y= x^2 2 e^ -x + (e- 1 2 ) e^ -x \)
Related: Mathematics — Differential Equations · All PYQ Banks