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  "body": "\n## Example\n\nConsider the simplest [[Sturm–Liouville problem|7WV6B2502K]]: find $\\lambda$ and a nonzero $u \\in C^2[0, 1]$ with\n$$-u''(x) = \\lambda u(x), \\quad u(0) = u(1) = 0.$$\nThe [[Green's function|SH6JY9RN2E]] of $-u''$ with these boundary conditions is\n$$G(x, s) = \\begin{cases} s(1 - x), \u0026 0 \\le s \\le x \\le 1, \\\\ x(1 - s), \u0026 0 \\le x \\le s \\le 1. \\end{cases}$$\nFor continuous $f$, the unique solution of $-u'' = f$, $u(0) = u(1) = 0$ is $u(x) = \\int_0^1 G(x, s) f(s) \\, ds$.\nTaking $f = \\lambda u$, the boundary value problem becomes the homogeneous Fredholm equation of the second kind\n$$u(x) = \\lambda \\int_0^1 G(x, s) \\, u(s) \\, ds.$$\nConversely, a continuous solution of this integral equation is $C^2$ and solves the boundary value problem.\n\nCheck: the kernel $G$ is continuous and [[symmetric|ADTYF8FG6C]].\nFor $u_n(x) = \\sin(n \\pi x)$, the function $v = u_n / (n^2 \\pi^2)$ satisfies $-v'' = u_n$ and $v(0) = v(1) = 0$.\nSo $\\int_0^1 G(x, s) \\sin(n \\pi s) \\, ds = \\sin(n \\pi x) / (n^2 \\pi^2)$, and the [[characteristic values|DC7HMHVREP]] of $G$ are $\\lambda_n = n^2 \\pi^2$, $n \\ge 1$.\nThese are exactly the eigenvalues of the boundary value problem.\n\n## Remarks\n\nSince $G$ is symmetric, the Hilbert–Schmidt theorem gives the expansion of every function $\\int_0^1 G(x, s) f(s) \\, ds$ in the sine series $\\sum_n c_n \\sin(n \\pi x)$.\nThe [[trace|3JVT83K8A1]] of $G$ is $\\int_0^1 x(1 - x) \\, dx = 1/6$.\nIt equals $\\sum_{n \\ge 1} 1/\\lambda_n = \\sum_{n \\ge 1} 1/(n^2 \\pi^2)$, which is Euler's identity $\\sum_{n \\ge 1} 1/n^2 = \\pi^2/6$.\n\n## Explanations\n\n### L1 human\n\nA differential equation with boundary conditions can be turned around.\nInstead of asking which $u$ has $-u''$ proportional to $u$, one asks which $u$ is reproduced, up to a factor, by integrating against $G$.\nThe Green's function $G$ is a tent-shaped kernel that undoes $-d^2/dx^2$.\nThe two problems have the same solutions, and the integral form brings the tools of compact symmetric operators: real eigenvalues, orthogonal eigenfunctions and expansions.\n",
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