Let y = f(x) be the real valued function defined on the interval (0, ) , satisfying y(1) = 0 and the differential equation x dy dx = y - x^3 . Then which of the following statements is (are) TRUE ?

Step-by-step solution

The given differential equation is x dy dx = y - x^3 . Dividing by x , we get: dy dx - 1 x y = -x^2 This is a linear differential equation of the form dy dx + P(x)y = Q(x) , where P(x) = - 1 x and Q(x) = -x^2 . Integrating Factor (IF) = e^ - 1 x dx = e^ - x = 1 x . Multiplying the differential equation by the IF: d dx ( y 1 x ) = -x Integrating both sides with respect to x : y x = - x^2 2 + C y = - x^3 2 + Cx Given y(1) = 0 : 0 = - 1 2 + C C = 1 2 Thus, f(x) = x 2 - x^3 2 . To find local extrema, we find f'(x) : f'