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  "body": "\n## Summary\n\nAn integral equation of the third kind is a [[linear integral equation|AYZCKG34QK]] $a_0(x) \\varphi(x) = f(x) + \\lambda \\int K(x, s) \\varphi(s) \\, ds$ whose coefficient $a_0$ vanishes at some points but not everywhere.\nIt sits between the [[first kind|ZDS77Q0F06]], where $a_0 = 0$, and the [[second kind|8KJCN49V64]], where $a_0 = 1$.\nNear the zeros of $a_0$ the equation behaves like one of the first kind, so its solutions may be singular there.\nSuch equations occur in transport theory and in some problems of mathematical physics.\n\n## Explanations\n\n### L1 human\n\nThe unknown appears outside the integral with a weight $a_0(x)$.\nWhere the weight is not zero one can divide by it and get an ordinary equation of the second kind.\nAt the points where the weight is zero this division fails, and the solution can blow up or need not exist.\nFor example, $x \\varphi(x) = f(x) + \\int_{-1}^{1} \\varphi(s) \\, ds$ on $[-1, 1]$ is of the third kind, since $x$ vanishes at $0$.\n",
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