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  "body": "\n## Summary\n\nAn integral equation of the second kind is a [[linear integral equation|AYZCKG34QK]] of the form $\\varphi(x) = f(x) + \\lambda \\int K(x, s) \\varphi(s) \\, ds$, where the unknown also appears outside the integral.\nIn operator form it reads $(I - \\lambda K) \\varphi = f$.\nFor reasonable kernels this equation is well behaved: Fredholm theory shows that it is uniquely solvable for all $f$ unless $\\lambda$ is a characteristic value.\nFor small $\\abs{\\lambda}$ the solution is given by the [[Liouville–Neumann series|H1HWSWME6B]].\n\n## Explanations\n\n### L1 human\n\nThe unknown equals a known function plus a correction that depends on the unknown itself.\nWhen the correction is small, one can solve the equation by starting with $f$ and correcting again and again.\nFor example, $\\varphi(x) = 1 + \\int_0^x \\varphi(s) \\, ds$ is of the second kind.\nStarting with $1$ and correcting gives $1$, $1 + x$, $1 + x + x^2/2$, and so on, which builds up $e^x$.\n",
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