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  "body": "\n## Statement\n\nLet $I \\subseteq \\R$ be an interval and let $p, q \\colon I \\to \\R$ be continuous.\nLet $y_1$ and $y_2$ be linearly independent real solutions of the [[linear differential equation|0QQRP17Z7S]]\n$$y'' + p(x) y' + q(x) y = 0 \\quad \\text{on } I.$$\nThen the zeros of $y_1$ and $y_2$ in $I$ strictly alternate.\nThat is, $y_1$ and $y_2$ have no common zero.\nMoreover, if $x_1 \u003c x_2$ are two consecutive zeros of $y_1$ in $I$, then $y_2$ has exactly one zero in the open interval $(x_1, x_2)$.\n\n## Remarks\n\nThe [[Wronskian|0KC1NMG90B]] $W = y_1 y_2' - y_1' y_2$ of two independent solutions never vanishes, so it has a constant sign on $I$.\nA common zero would make $W$ vanish there.\nIf $y_2$ had no zero in $(x_1, x_2)$, the function $y_1 / y_2$ would be differentiable on $[x_1, x_2]$ with derivative $-W / y_2^2$ of constant sign.\nYet it vanishes at $x_1$ and $x_2$, which contradicts Rolle's theorem.\nApplying the same argument with the roles exchanged shows that there is at most one zero.\nExample: for $y'' + y = 0$, the solutions $\\sin x$ and $\\cos x$ have zeros at $k\\pi$ and at $\\pi/2 + k\\pi$, which alternate.\nLinear independence is needed: $y_1 = \\sin x$ and $y_2 = 2 \\sin x$ have the same zeros.\n\n## Explanations\n\n### L1 human\n\nTake two genuinely different solutions of a second-order linear equation.\nBetween two crossings of the axis by one of them, the other crosses the axis exactly once.\nSo the zeros of the two solutions interlace like the zeros of sine and cosine.\n\n### L2 human\n\nThe theorem shows that oscillation is a property of the equation and not of the solution.\nIf one nonzero solution has infinitely many zeros on an interval, then every nonzero solution does.\nIt also shows that the numbers of zeros of two solutions on an interval differ by at most one.\nFor equations $y'' + q y = 0$ it also follows from the [[Sturm comparison theorem|W7VXNYZQN2]] with $q_1 = q_2$, and it is a basic tool in Sturm–Liouville theory.\n",
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