Let f: R R be such that f(xy) = f(x)f(y) , for all x, y R and f(0) 0 . Let g: [1, ) R be a differentiable function such that x^2 g(x) = ₁^x (t^2 f(t) - tg(t)) ,dt . Then g(2) is equal to :

Step-by-step solution

Given f(xy) = f(x)f(y) for all x, y R and f(0) 0 . Substituting y = 0 , we get f(0) = f(x)f(0) . Since f(0) 0 , dividing by f(0) gives f(x) = 1 for all x R . The given integral equation is: x^2 g(x) = ₁^x (t^2 f(t) - tg(t)) ,dt Substituting f(t) = 1 : x^2 g(x) = ₁^x (t^2 - tg(t)) ,dt Differentiating both sides with respect to x using Leibniz's rule: 2x g(x) + x^2 g'(x) = x^2 - xg(x) x^2 g'(x) + 3xg(x) = x^2 Dividing by x (since x 1 ): x g'(x) + 3g(x) = x g'(x) + 3 x g(x) = 1 This is a linear differential equation.