Counterexample

Is this true?

Every integral equation of the second kind has a unique solution

No — it is false.

The counterexample

$y(x) - 3\int_{0}^{1}$ tx y(t)dt = eˣ

Why it fails

$\lambda = 3$ is an eigenvalue of the kernel tx, and eˣ is not orthogonal to t — no solution exists.

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Execution slip

The idea was right. The computation was not.

Volterra equations have eigenvalues just like Fredholm equations

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integral equations

Go deeper· Explained in full in Integral equations: Fredholm vs Volterra· Asked in CSIR NET June 2025· From Linear Integral Equations › Fredholm and Volterra equations

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.

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