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  "body": "\n## Summary\n\nThe method of undetermined coefficients finds a particular solution of a [[linear equation with constant coefficients|0A6PGTTP2A]] $L y = b(x)$ when $b$ has a special form.\nThe right side must be a sum of terms $x^k e^{\\lambda x}$, $x^k e^{\\lambda x} \\cos(\\omega x)$ and $x^k e^{\\lambda x} \\sin(\\omega x)$.\nFor such a term, one tries a function of the same type, with a general polynomial of degree $k$ and unknown coefficients. If $\\lambda$ (or $\\lambda + i\\omega$) is a root of multiplicity $s$ of the [[characteristic polynomial|0QDXMG46QW]], one multiplies the trial function by $x^s$.\nSubstitution into the equation gives linear equations for the unknown coefficients.\n\n## Remarks\n\nExample: for $y'' + y = x$, try $y_p = A x + B$. Then $A x + B = x$, so $y_p = x$.\nExample with resonance: for $y'' + y = \\cos x$, the number $i$ is a simple root of $r^2 + 1$. So try $y_p = x (A \\cos x + B \\sin x)$, which gives $y_p = \\frac{1}{2} x \\sin x$.\nThe method works because the derivatives of these functions stay in a finite-dimensional space. For other right sides, use [[variation of parameters|0Y6TH5PQVS]].\n\n## Explanations\n\n### L1 human\n\nGuess that the answer looks like the right side, with unknown numbers in it.\nPut the guess into the equation and match the coefficients.\nIf the guess already solves the homogeneous equation, multiply it by $x$.\n",
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