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

by RoRi

January 30, 2016

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 $. \qquad \blacksquare$

Stumbling Robot