Let y ( x ) be the solution of the differential equation ( x log x ) d y d x + y = 2 x log x ,   ( x ≥ 1 ) . Then y ( e ) is equal to

Step-by-step solution

Given, ( x log x ) d y d x + y = 2 x log x ⇒ d y d x + 1 x log x y = 2 Integrating factor I . F = e ∫ P d x = e ∫ 1 x log x d x = e ∫ 1 / x log x d x = e log log x = log x The solution of the equation becomes y · e ∫ P d x = ∫ Q · e ∫ P d x d x + c , where, c is the constant of integration. ⇒ y log x = ∫ 2 log x d x + c ⇒ y log x = 2 x log x - x + c Let, P 1 , y 1 be any point on the curve. ⇒ y 1 · 0 = 2 0 - 1 + c ⇒ c = 2 ⇒