If the curve y = f(x) passes through the point (1, e) and satisfies the differential equation dy = y(2 + _e x) ,dx , x > 0 , then f(e) is equal to :

Step-by-step solution

Given differential equation is dy y = (2 + _e x) dx Integrating both sides, we get: dy y = (2 + _e x) dx _e y = 2x + x _e x - x + C _e y = x + x _e x + C Since the curve passes through (1, e) , substitute x = 1 and y = e : _e e = 1 + 1 _e 1 + C 1 = 1 + 0 + C C = 0 The equation of the curve is _e y = x + x _e x To find f(e) , substitute x = e : _e y = e + e _e e _e y = e + e(1) = 2e y = e^ 2e Answer: e^ 2e