Let f(x) be a differentiable function defined on (0, ) such that f(x) > 0 and f(x) = 1 + ₁^ x ( f(t) t - t^2 (f(t))^2 ) dt for all x > 0 . Then the value of e^6 (f(e⁻²))^3 is equal to

Step-by-step solution

Differentiating the given integral equation with respect to x using the Leibniz rule, we get f'(x) = f(x) x - x^2 (f(x))^2 f'(x) - 1 x f(x) = -x^2 (f(x))⁻² This is a Bernoulli differential equation. Multiplying both sides by 3(f(x))^2 , we obtain 3(f(x))^2 f'(x) - 3 x (f(x))^3 = -3x^2 Let v = (f(x))^3 . Then dv dx = 3(f(x))^2 f'(x) . The equation becomes a linear differential equation: dv dx - 3 x v = -3x^2 The integrating factor (I.F.) is e^ - 3 x dx = e^ -3 x = x⁻³ . Multiplying the equation by the I.F. and integ