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  "body": "\n## Definition\n\nLet $a \u003c b$ be real numbers and consider a [[linear differential equation|0QQRP17Z7S]] of order $2$ on $[a, b]$.\nA function $y \\in C^1([a, b])$ satisfies the periodic boundary conditions on $[a, b]$ if\n$$y(a) = y(b) \\quad \\text{and} \\quad y'(a) = y'(b).$$\nFor an equation of order $n$, the periodic boundary conditions are $y^{(k)}(a) = y^{(k)}(b)$ for $k = 0, 1, \\ldots, n - 1$.\n\n## Remarks\n\nExample: on $[0, 2\\pi]$, the function $y(x) = \\sin x$ satisfies $y(0) = 0 = y(2\\pi)$ and $y'(0) = 1 = y'(2\\pi)$.\nSo it satisfies the periodic boundary conditions, and it solves $y'' + y = 0$.\nNon-example: on $[0, \\pi]$, the same function has $y(0) = y(\\pi) = 0$ but $y'(0) = 1 \\ne -1 = y'(\\pi)$, so the second condition fails.\nThe eigenvalue problem $-y'' = \\lambda y$ with periodic boundary conditions on $[0, 2\\pi]$ has the eigenvalues $\\lambda = k^2$ for $k = 0, 1, 2, \\ldots$.\nThe eigenvalue $0$ has the eigenfunction $1$, and each $k^2$ with $k \\ge 1$ has the two independent eigenfunctions $\\cos kx$ and $\\sin kx$.\nUnlike separated conditions, periodic conditions couple the two endpoints, so eigenvalues of multiplicity $2$ can occur.\nIf the coefficients of the equation are $(b - a)$-periodic, a solution with these conditions extends to a $(b - a)$-periodic solution on $\\R$.\n\n## Explanations\n\n### L1 human\n\nPeriodic boundary conditions say that the start and the end of the interval look the same to the solution.\nThe value and the slope must match at both ends.\nThen copies of the solution glue together into a smooth repeating function on the whole line.\n",
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