The solution of the differential equation d x d y = x 2 e y - x ∀ x > 0 is λ x + 2 c x 2 e y = e y (where, c is an arbitrary constant). Then, λ is equal to

Step-by-step solution

The given equation is e − y d y d x + e − y ⋅ 1 x = 1 x 2 Let, - e - y = t ⇒ e - y d y d x = d t d x Thus we have, d t d x - t 1 x = 1 x 2 Integrating factor = e - ∫ 1 x d x = e - ln ⁡ x = 1 x Thus, the solution is t ⋅ 1 x = ∫ 1 x 2 ⋅ 1 x d x ⇒ - e y x = x - 2 - 2 + c ⇒ - 2 x e - y = - 1 + 2 c x 2 or 2 x e - y = 1 - 2 c x 2 Hence, λ = 2