The solution of the differential equation d y d x + x y ln ⁡ y = x 3 y is equal to (where, C is the constant of integration)

Step-by-step solution

1 y d y d x + x ln ⁡ y = x 3 Let ln ⁡ y = t 1 y d y d x = d t d x d t d x + t x = x 3 I.F. = e ∫ x d x = e x 2 2 t e x 2 2 = ∫ e x 2 2 x 3 d x ln ⁡ y e x 2 2 = ∫ e x 2 2 ⋅ x x 2 d x ln ⁡ y e x 2 2 = e x 2 2 ⋅ x 2 - ∫ 2 x e x 2 2 d x ln ⁡ e x 2 2 = e x 2 2 x 2 - 2 e x 2 2 + C ln ⁡ y ⋅ e x 2 2 = e x 2 2 x 2 - 2 + C ln ⁡ y = x 2 - 2 + C e - x 2 2