The equation of the curve satisfying the differential equation d y d x + 2 y x 2 = 2 x 2 and passing through 1 2 , e 4 + 1 is

Step-by-step solution

Given equation is a linear differential equation Integrating factor = e ∫ 2 x 2 d x = e − 2 x ∴ The solution is y e − 2 x = ∫ 2 e − 2 x x 2 d x or y e − 2 x = ∫ e t d t (Putting - 2 x = t ) ⇒ y e − 2 x = e t + c ⇒ y e − 2 x = e − 2 x + c ⇒ y = 1 + c e 2 x At x = 1 2 , y = e 4 + 1 , ⇒ c = 1 ∴ The equation of the curve is y = 1 + e 2 x