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  "body": "\n## Definition\n\nLet $t_0 \\in \\R$ and let $q \\colon [t_0, \\infty) \\to \\R$ be continuous.\nConsider the [[linear differential equation|0QQRP17Z7S]]\n$$y'' + q(t) y = 0 \\quad \\text{on } [t_0, \\infty).$$\nA real [[solution|07VX34QXAV]] $y$ that is not identically zero is oscillatory if it has arbitrarily large zeros.\nThat is, for each $T \\ge t_0$ there is $t \u003e T$ with $y(t) = 0$.\nOtherwise $y$ is nonoscillatory.\nThe equation is oscillatory if it has an oscillatory solution, and nonoscillatory otherwise.\nOscillation theory is the study of the zeros of solutions of such equations and their generalisations.\nIt looks in particular for criteria for oscillation in terms of the coefficients.\n\n## Remarks\n\nBy the Sturm separation theorem, if one nonzero solution is oscillatory then every nonzero solution is oscillatory.\nSo the equation is oscillatory exactly when all its nonzero solutions are.\nExample: $y'' + y = 0$ is oscillatory, since $y = \\sin t$ vanishes at $t = k\\pi$ for every integer $k$.\nNon-example: for $y'' = 0$ every nonzero solution $y = A t + B$ has at most one zero, so the equation is nonoscillatory.\nThe Euler equation $y'' + \\frac{c}{t^2} y = 0$ on $[1, \\infty)$ is oscillatory exactly when $c \u003e \\frac{1}{4}$.\nA classical criterion says that the equation is oscillatory if $\\int_{t_0}^\\infty q(t) \\, dt = +\\infty$.\n\n## Explanations\n\n### L1 human\n\nA solution oscillates if its graph keeps crossing the axis forever.\nOscillation theory looks for conditions on the equation that force, or forbid, this behaviour.\n",
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