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Prove that there is a unique function satisfying a given integral equation

Prove there exists a unique function f(x) which is continuous on \mathbb{R}_{>0} and that satisfies the equation

    \[ f(x) = 1 + \frac{1}{x} \int_1^x f(t) \, dt \]

for all positive x and find the function.


Proof. Let

    \[ y = \int_1^x f(t) \, dt. \]

Then we are looking for solutions to the differential equation

    \[ y' - \frac{1}{x} y = 1. \]

We apply Theorem 8.3 (page 310 of Apostol) with

    \[ P(x) = -\frac{1}{x}, \qquad Q(x) = 1, \qquad a = 1, \ b = 0. \]

This gives us

    \[ A(x) = \int_1^x -\frac{1}{t} \, dt = -\log x. \]

Therefore,

    \begin{align*}  y &= b e^{\log x} + e^{\log x} \int_1^x e^{-\log t} \,dt  \\  &= x \int_1^x \frac{1}{t} \, dt \\  &= x \log x. \end{align*}

Therefore,

    \[ y' = f(x) = \log x + 1. \]

By the uniqueness property of Theorem 8.3, we know this solution is unique. \qquad \blacksquare

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