Let f: (0, ) R be a differentiable function satisfying f(x) = k x^3 + ₀^1 x^2 f( x) d , where k is a constant. If f(2) = 24(e^2 - 1) , then the value of k is equal to _______.

Step-by-step solution

Given, f(x) = k x^3 + ₀^1 x^2 f( x) d Let t = x dt = x d d = dt x When =0, t=0 and when =1, t=x . ₀^1 x^2 f( x) d = ₀^x x^2 f(t) dt x = x ₀^x f(t) dt Substituting this back into the given equation: f(x) = k x^3 + x ₀^x f(t) dt Dividing by x (since x > 0 ): f(x) x = k x^2 + ₀^x f(t) dt Differentiating both sides with respect to x using the quotient rule and Leibniz rule: x f'(x) - f(x) x^2 = 2k x + f(x) x f'(x) - f(x) = 2k x^3 + x^2 f(x) x f'(x) - (1 + x^2) f(x) = 2k x^3 Dividing by x : f'(x) - ( 1 x + x ) f(x) = 2k