Let x = f(y) be the solution of the differential equation y y , dx - (x + y^2 ( y)^2) , dy = 0 for y > 1 , satisfying the condition f(e) = e^2 2 . The area of the region bounded by the curve x = f(y) , the y -axis, and the lines y = 1 and y = e is :

Step-by-step solution

The given differential equation can be rewritten as : y y dx dy - x = y^2 ( y)^2 dx dy - 1 y y x = y y This is a linear differential equation in x . The integrating factor (I.F.) is : I.F. = e^ - 1 y y dy = e^ - ( y) = 1 y Multiplying by the I.F., we get : d dy ( x y ) = y Integrating both sides with respect to y : x y = y^2 2 + C Using the initial condition f(e) = e^2 2 , we substitute x = e^2 2 and y = e : e^2 2 e = e^2 2 + C e^2 2 = e^2 2 + C C = 0 So, the equation of the curve is x = y^2 2 y . The area bounded