A curve x = x(y) passes through the origin and satisfies the differential equation dy = dx x y + 2y . Then the value of x ( 2 ) is equal to

Step-by-step solution

Given differential equation: dy = dx x y + 2y Taking the reciprocal, we get: dx dy = x y + 2y Rearranging into the standard form of a linear differential equation: dx dy - x y = 2y Here, P(y) = - y and Q(y) = 2y . Integrating Factor (I.F.) = e^ - y dy = e^ - y The general solution is given by: x e^ - y = 2y e^ - y dy + C To solve the integral, let t = - y dt = - y dy . (2 y y) e^ - y dy = 2(-t) e^t (-dt) = 2t e^t dt Using integration by parts: 2t e^t dt = 2(t e^t - e^t) = 2(- y - 1)e^ - y Substituting this back: x