Let f: (0, ) R be a differentiable function satisfying f(1) = e and ₁^ x (t+1) f(t) dt = x^2 f(x) - e for all x > 0 . Then the value of f(2) is equal to

Step-by-step solution

₁^ x (t+1) f(t) dt = x^2 f(x) - e Differentiating both sides with respect to x using the Newton-Leibniz formula and the product rule: (x+1) f(x) = 2x f(x) + x^2 f'(x) Rearranging the terms to isolate f'(x) : x^2 f'(x) = (x+1 - 2x) f(x) x^2 f'(x) = (1-x) f(x) Separating the variables: f'(x) f(x) = 1-x x^2 = 1 x^2 - 1 x Integrating both sides with respect to x : |f(x)| = - 1 x - x + C Using the given initial condition f(1) = e : e = - 1 1 - 1 + C 1 = -1 - 0 + C C = 2 Thus, the function is given by: |f(x)| = - 1 x - x