Consider the differential equation, y 2 d x + x - 1 y d y = 0 . If value of y is 1 when x = 1 , then the value of x for which y = 2 , is

Step-by-step solution

Given differential equation is y 2 d x = 1 y - x d y ⇒ y 2 d x d y + x = 1 y ⇒ d x d y + 1 y 2 x = 1 y 3 It is a linear differential equation whose integrating factor I . F .   = e ∫ d y y 2 = e - 1 y Solution of a given differential equation can be written as x e - 1 y = ∫ e - 1 y 1 y 3 d y = I Let - 1 y = t ⇒ d y y 2 = d t ⇒ I = ∫ - t e t   d t = e t 1 - t + C , (Integrating by parts) ⇒   Solution of differential equation is x e - 1 y = e - 1 y 1 + 1