Let f: [0, ) R be a differentiable function satisfying f(x) = x^2 + x^2 ₀^1 t f(tx) dt . The value of f( 2 5 ) is equal to _______.

Step-by-step solution

Given, f(x) = x^2 + x^2 ₀^1 t f(tx) dt Let u = tx du = x dt dt = du x When t=0, u=0 and when t=1, u=x . ₀^1 t f(tx) dt = ₀^x u x f(u) du x = 1 x^2 ₀^x u f(u) du Substituting this back into the given equation: f(x) = x^2 + x^2 ( 1 x^2 ₀^x u f(u) du ) f(x) = x^2 + ₀^x u f(u) du Differentiating both sides with respect to x using Leibniz rule: f'(x) = 2x + x f(x) f'(x) - x f(x) = 2x This is a linear differential equation of the first order. The integrating factor (I.F.) is: I.F. = e^ -x dx = e^ -x^2/2 Multiplying both