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  "body": "\n## Summary\n\nA homogeneous integral equation is a [[linear integral equation|AYZCKG34QK]] whose free term $f$ is zero, such as $\\varphi(x) = \\lambda \\int_a^b K(x, s) \\varphi(s) \\, ds$.\nIt always has the zero solution, and its solutions form a vector space.\nThe values of $\\lambda$ for which it has a nonzero solution are the characteristic values of the kernel.\nBy the Fredholm alternative, these values decide whether the inhomogeneous equation is uniquely solvable.\n\n## Explanations\n\n### L1 human\n\nHomogeneous means that the equation has no term without the unknown.\nIt is the integral version of a system $A x = 0$ of linear equations.\nZero is always a solution; the interesting question is whether there are others.\nFor example, $\\varphi(x) = \\lambda \\int_0^1 \\varphi(s) \\, ds$ has the nonzero solution $\\varphi = 1$ exactly when $\\lambda = 1$.\n",
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