Let y = y ( x ) be the solution of the differential equation 3 y 2 - 5 x 2 y d x + 2 x x 2 - y 2 d y = 0 such that y ( 1 ) = 1 . Then ( y ( 2 ) ) 3 - 12 y ( 2 ) is equal to :

Step-by-step solution

Given, 3 y 2 - 5 x 2 y d x + 2 x x 2 - y 2 d y = 0 ⇒ d y d x = y 5 x 2 - 3 y 2 2 x x 2 - y 2 This is homogeneous differential equation Let y = t x ⇒ d y d x = t + x d t d x t + x d t d x = ( t x ) 5 x 2 - 3 t 2 x 2 2 x x 2 - t 2 x 2 = t 5 - 3 t 2 2 1 - t 2 ⇒ x d t d x = 5 t - 3 t 3 2 - 2 t 2 - t = 3 t - t 3 2 - 2 t 2 ⇒ 2 1 - t 2 3 t - t 3 d t = 1 x d x Taking integration on both sides we get, ∫ 2 1 - t 2 3 t - t 3 d t = ∫ 1 x d x . . . . . . 1 Now let 3 t - t 3 = z ⇒ 3 1 -