Let the solution curve y = y x of the differential equation 1 + e 2 x d y d x + y = 1 pass through the point 0 , π 2 . Then, lim x → ∞ e x y x is equal to

Step-by-step solution

Given, 1 + e 2 x d y d x + y = 1 Now on rearranging we get, ⇒ d y d x + y = 1 1 + e 2 x We can see it is a linear differential equation, So integrating factor is e ∫ 1 · d x = e x So solution will be y · I F = ∫ 1 1 + e 2 x × I F   d x ⇒ y e x = ∫ 1 1 + e 2 x × e x   d x ⇒ y · e x = tan - 1 e x + c Now as curve is passing through 0 , π 2 so ⇒ c = π 4 Now calculating the limit lim x → ∞ y · e x we get, ⇒ lim