{
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  "body": "\n## Summary\n\nVariation of parameters is a method to find a particular solution of an inhomogeneous linear equation from a [[fundamental system|0H4KNNCHPY]] of the homogeneous equation.\nLet $p, q, f$ be continuous on an open interval $I$. Let $(y_1, y_2)$ be a fundamental system of $y'' + p y' + q y = 0$ on $I$, with [[Wronskian|0N5NF1RGBP]] $W$.\nOne looks for $y = u_1 y_1 + u_2 y_2$ with the extra condition $u_1' y_1 + u_2' y_2 = 0$. Then the equation $y'' + p y' + q y = f$ becomes $u_1' y_1' + u_2' y_2' = f$.\nThis linear system gives $u_1' = -y_2 f / W$ and $u_2' = y_1 f / W$, so a particular solution is\n$$y_p = -y_1 \\int \\frac{y_2 f}{W}\\, dx + y_2 \\int \\frac{y_1 f}{W}\\, dx.$$\n\n## Remarks\n\nThe method works for each continuous right side, unlike the [[method of undetermined coefficients|0XRB469Z32]].\nExample: for $y'' + y = \\tan x$ on $(-\\pi/2, \\pi/2)$, with $y_1 = \\cos x$, $y_2 = \\sin x$ and $W = 1$, one gets $y_p = -\\cos x \\ln\\left(\\sec x + \\tan x\\right)$.\nFor first-order systems, the same idea gives the [[variation of constants formula|0S1KG0XE40]].\n\n## Explanations\n\n### L1 human\n\nThe general solution of the homogeneous equation is $c_1 y_1 + c_2 y_2$ with constants.\nThe idea is to let the constants vary with $x$.\nA smart extra condition makes the equations for $u_1'$ and $u_2'$ simple.\n",
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