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  "body": "\n## Summary\n\nSturm–Liouville theory studies the eigenvalue problems $-(p y')' + q y = \\lambda w y$ on an interval, with boundary conditions at the ends.\nFor a regular problem with separated boundary conditions, the eigenvalues are real and simple and form a sequence $\\lambda_1 \u003c \\lambda_2 \u003c \\cdots$ that tends to $+\\infty$.\nThe $n$-th eigenfunction has exactly $n - 1$ zeros inside the interval, and the eigenfunctions form an orthogonal basis of the weighted space $L^2((a, b), w)$.\nThe theory explains Fourier series and many classical special functions, and it is the model case of the spectral theory of [[self-adjoint operators|0NEGN5CRSC]].\n\n## Explanations\n\n### L1 human\n\nA vibrating string of varying thickness can only vibrate at certain frequencies.\nSturm–Liouville theory finds these frequencies as the [[eigenvalues|0BQAA2BK3Q]] of a second-order differential equation with conditions at the two ends of the string.\nFor a uniform string on $[0, \\pi]$ fixed at both ends, the equation is $-y'' = \\lambda y$ with $y(0) = y(\\pi) = 0$.\nThe eigenvalues are $1, 4, 9, \\ldots$ with the shapes $\\sin x, \\sin 2x, \\sin 3x, \\ldots$.\nThe $n$-th shape crosses zero $n - 1$ times between the ends, and the shapes are building blocks for any other shape.\n",
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