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  "body": "\n## Definition\n\nLet $a \u003c b$, let $a_0, f \\colon [a, b] \\to \\C$ and a [[kernel|GSMBXT7FPK]] $K \\colon [a, b] \\times [a, b] \\to \\C$ be given, and let $\\lambda \\in \\C$.\nFor each $x \\in [a, b]$, let $I_x$ be either $[a, b]$ or $[a, x]$.\nA [[linear integral equation|AYZCKG34QK]]\n$$a_0(x) \\varphi(x) = f(x) + \\lambda \\int_{I_x} K(x, s) \\varphi(s) \\, ds \\quad \\text{for all } x \\in [a, b]$$\nis *homogeneous* if $f(x) = 0$ for all $x \\in [a, b]$, and *inhomogeneous* otherwise.\nFor an equation of the [[second kind|8KJCN49V64]], the homogeneous equation is\n$$\\varphi(x) = \\lambda \\int_{I_x} K(x, s) \\varphi(s) \\, ds \\quad \\text{for all } x \\in [a, b].$$\n\n## Remarks\n\nExample: $\\varphi(x) = \\lambda \\int_0^1 x s \\, \\varphi(s) \\, ds$ on $[0, 1]$ is homogeneous.\nEvery solution has the form $\\varphi(x) = c x$ with $c = \\lambda c / 3$, so there is a nonzero solution exactly when $\\lambda = 3$.\n\nNon-example: $\\varphi(x) = x + \\lambda \\int_0^1 x s \\, \\varphi(s) \\, ds$ is inhomogeneous, since its free term $f(x) = x$ is not zero.\n\nThe solutions of a homogeneous equation form a vector space, like the solutions of a [[homogeneous system of linear equations|04KGXJG1A2]].\nThe difference of two solutions of an inhomogeneous equation solves the homogeneous one.\n",
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