A continuous function f: R - R satisfies the equation f(x) = (1 + x²) * [1 + integral from 0 to x of f(t)² / (1 + t²) dt] Find the value of (17/15) * f(-2).

Question

A continuous function f: R -> R satisfies the equation f(x) = (1 + x²) * [1 + integral from 0 to x of f(t)² / (1 + t²) dt] Find the value of (17/15) * f(-2).

Accepted Answer

1. Differentiating the integral equation: f'(x) = 2x(1 + integral) + f(x)²/(1+x²) * (1+x²) = 2x * f(x)/(1+x²) + f(x)². This is a Bernoulli ODE. Let g = 1/f: -g' = 2x*g/(1+x²) + 1 * f/f²... After substitution: g' + 2xg/(1+x²) = -1. Integrating factor = e^(integral 2x/(1+x²) dx) = 1+x². So d/dx[(1+x²)g] = -(1+x²). Integrate: (1+x²)g = -(x + x³/3) + C. Initial condition: f(0) = 1 => g(0) = 1 => C = 1. So g(x) = (1 - x - x³/3)/(1+x²). Then f(-2) = 1/g(-2) = (1+4)/(1 - (-2) - (-8)/3) = 5/(3 + 8/3) = 5/(17/3) = 15/17. Finally (17/15)*f(-2) = (17/15)*(15/17) = 1.

A continuous function f: R -> R satisfies the equation f(x) = (1 + x²) * [1 + integral from 0 to x of f(t)² / (1 + t²) dt] Find the value of (17/15) * f(-2).

Solution

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