If y = f x satisfies the differential equation d y d x + y = sin ⁡ x + cos ⁡ x and f 0 = 2 , then lim x → ∞ ⁡ f x - sin ⁡ x is equal to

Step-by-step solution

Given equation is a linear differential equation Integrating factor = e ∫ d x = e x So the equation is y e x = ∫ e x sin ⁡ x + cos ⁡ x d x = e x sin ⁡ x + C ⇒ f x = sin ⁡ x + C e - x Since f 0 = 2 ⇒ C = 2 Hence, f x = sin ⁡ x + 2 e - x ⇒ f x - sin ⁡ x = 2 e - x Therefore, lim x → ∞ ⁡ f x - sin ⁡ x = 0