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  "body": "\n## Summary\n\nOscillation theory studies the zeros of solutions of differential equations, above all of second-order linear equations $(p y')' + q y = 0$.\nIts main questions are whether solutions have infinitely many zeros, how the zeros are spaced, and how the zeros of different solutions are related.\nThe theory started with the comparison and separation theorems of Charles Sturm.\nIt gives criteria in terms of the coefficients, and it is linked to [[Sturm–Liouville theory|7WV6B2502K]], where the $n$-th eigenfunction has $n - 1$ zeros.\n\n## Explanations\n\n### L1 human\n\nA solution oscillates if it crosses the axis again and again, without end, like a sine wave.\nOscillation theory asks which equations force their solutions to behave like this.\nThe equation $y'' + y = 0$ is oscillatory, since its solutions $A \\sin(x + c)$ have infinitely many zeros.\nThe equation $y'' - y = 0$ is not: its solutions $A e^x + B e^{-x}$ have at most one zero.\nFor $y'' + q(x) y = 0$, a large positive $q$ means many oscillations, a phenomenon the Sturm comparison theorem makes precise.\n",
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