Let y = y x be the solution of the differential equation d y d x + 2 y = f x , where f x = 1 , x ∈ 0 , 1 0 , o t h e r w i s e . If y ( 0 ) = 0 , then y 3 2 is

Step-by-step solution

d y d x + 2 y = f x is a linear differential equation. Integrating Factor = e ∫ 2 d x = e 2 x Solution of the above equation is y .   e 2 x = ∫ f x . e 2 x d x + C y x = e - 2 x ∫ 0 x f x . e 2 x d x + C e - 2 x y 0 = 0 ⇒ C = 0 ⇒ y x = e - 2 x ∫ 0 x f x . e 2 x d x ∴ y 3 2 = e - 3 ∫ 0 1 e 2 x d x + ∫ 1 3 / 2 0 . d x = e - 3 2 e 2 - 1 = e 2 - 1 2 e 3