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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

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

The kind of mistake this is

Execution slip

The idea was right. The computation was not.

Drill statements like this

integral equations

Others that fail the same way

Volterra equations have eigenvalues just like Fredholm equations

From Linear Integral Equations › Fredholm and Volterra equations

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