The solution of the differential equation xdy = tan ⁡ y + e 1 / x 2 x sec ⁡ y dx is (where C is the constant of integration)

Step-by-step solution

d y d x = tan ⁡ y x + e 1 x 2 x 2 sec ⁡ y cos ⁡ y d y d x - sin ⁡ y x = e 1 x 2 x 2 Let, sin ⁡ y = t cos ⁡ y d y d x = d t d x ⇒ d t d x - t x = e 1 x 2 x 2 I.F. = e - ∫ 1 x d x = e ln ⁡ 1 x = 1 x ⇒ t x = ∫ e 1 x 2 x 3 d x ⇒ t x = - 1 2 ∫ - 2 x 3 e 1 x 2 d x ⇒ sin ⁡ y x = - 1 2 e 1 x 2 + C