Let the solution curve y = f x of the differential equation d y d x + x y x 2 - 1 = x 4 + 2 x 1 - x 2 , x ∈ - 1 , 1 pass through the origin. Then ∫ - 3 2 3 2 f x d x is equal to

Step-by-step solution

d y d x + x y x 2 - 1 = x 4 + 2 x 1 - x 2 is a linear differential equation. Here I . F . = e ∫ x x 2 - 1 d x = e 1 2 ∫ 2 x x 2 - 1 d x = e 1 2 ln x 2 - 1 i.e. I . F . = 1 - x 2 x (-1,1) So, solution of the differential equation is y · 1 - x 2 = ∫ x 4 + 2 x 1 - x 2 · 1 - x 2 d x y 1 - x 2 = ∫ x 4 + 2 x d x y 1 - x 2 = x 5 5 + x 2 + C Since the curve passes through origin so x = 0 ,   y = 0 ⇒ C = 0 i.e. y = x 5 5 1 - x 2 + x 2 1 - x 2 Now, ∫ - 3 2 3 2 f x d x = &#87