The solution curve of the differential equation, 1 + e - x 1 + y 2 d y d x = y 2 which passes through the point 0 , 1 , is

Step-by-step solution

1 + e - x 1 + y 2 d y d x = y 2 Separate variables ⇒ 1 + y 2 d y y 2 = d x 1 + e - x Integrate both side ⇒ ∫ 1 + y 2 d y y 2 = ∫ d x 1 + e - x ⇒ ∫ 1 + 1 y 2 d y = ∫ e x d x e x + 1 ⇒ y - 1 y = ln e x + 1 + C Since curve pass through 0 ,   1 . ⇒ 1 - 1 1 = ln e 0 + 1 + C   ⇒ C = - ln 2 ⇒ y - 1 y = ln e x + 1 - ln 2 ⇒ y 2 - 1 y = ln e x + 1 2 ⇒ y 2 = 1 + y log e 1 + e x 2