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

Step-by-step solution

Given differential equation is x 3 d y + x y - 1 d x = 0 ⇒ d y d x = 1 - x y x 3 ⇒ d y d x = 1 x 3 - y x 2 ⇒ d y d x + y x 2 = 1 x 3 This is a linear differential equation. If IF = e ∫ 1 x 2 d x = e - 1 x Solution of a given linear differential equation is y e - 1 x = ∫ e - 1 x 1 x 3 d x Put - 1 x = t ⇒ d x x 2 = d t ⇒ y e - 1 x = - ∫ e - t t d t ⇒ y e - 1 x = t e - t + e - t + C ⇒ y = t + 1 + C e 1 x ⇒ y = 1 x + 1 +   C e 1 x Put x = 1 2 , then