Let y = y x be the solution of the differential equation x 1 - x 2 d y d x + 3 x 2 y - y - 4 x 3 = 0 , x > 1 with y 2 = - 2 . Then y 3 is equal to

Step-by-step solution

Given x 1 - x 2 d y d x + 3 x 2 y - y - 4 x 3 = 0 x - x 3 d y d x + 3 x 2 - 1 y = 4 x 3 d y d x + 3 x 2 - 1 x - x 3 y = 4 x 3 x - x 3 This is a linear differential equation of the form d y d x + P y = Q Here I F = e ∫ P d x = e ∫ 3 x 2 - 1 x - x 3 d x Let x - x 3 = t ⇒ I F = e ∫ - dt t = e - ln t = 1 t ∴   I F = 1 x - x 3 So the general solution of the differential equation will be y × I F = ∫ Q × I F d x y 1 x - x 3 = ∫ 4 x 3 x - x 3 × 1 x - x 3 d x = &