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  "body": "\n## Definition\n\nLet $D = d/dt$ be the [[differentiation operator|BB15T3J891]], and let $L = a_n D^n + \\dots + a_1 D + a_0$ with $a_0, \\dots, a_n \\in \\C$ and $a_n \\ne 0$.\nLet $f$ be a function on $\\R$, and let $A(D) = b_m D^m + \\dots + b_0$ be a nonzero polynomial in $D$ with constant coefficients.\nIf $A(D) f = 0$, then $A$ is an *annihilator* of $f$.\nThe *annihilator method* for $L y = f$ consists of these steps.\n\n1. Find an annihilator $A$ of $f$.\n2. Solve the homogeneous equation $A(D) L(D) y = 0$ by the roots of the polynomial $A(r) L(r)$. A root $r$ of multiplicity $k$ gives $e^{rt}, t e^{rt}, \\dots, t^{k-1} e^{rt}$.\n3. Remove from this solution the terms that already solve $L y = 0$, and keep the remaining terms with unknown coefficients as a trial function $y_p$.\n4. Substitute $y_p$ into $L y = f$ and solve the resulting linear equations for the coefficients.\n\n## Remarks\n\nThe functions with an annihilator are exactly the finite sums of terms $t^k e^{\\alpha t}$, $t^k e^{\\alpha t} \\cos \\beta t$ and $t^k e^{\\alpha t} \\sin \\beta t$.\nFor instance, $t^k e^{\\alpha t}$ is annihilated by $(D - \\alpha)^{k+1}$, and $e^{\\alpha t} \\cos \\beta t$ by $(D - \\alpha)^2 + \\beta^2$.\nStep 4 always has a solution, which justifies the trial functions of the [[method of undetermined coefficients|0XRB469Z32]].\n\nExample: $y'' - y = e^t$.\nHere $L = D^2 - 1$ and $A = D - 1$, and $A L = (D - 1)^2 (D + 1)$ gives the solutions $e^t, t e^t, e^{-t}$.\nRemoving $e^t$ and $e^{-t}$, which solve $L y = 0$, leaves $y_p = c\\, t e^t$.\nThen $y_p'' - y_p = c (t + 2) e^t - c t e^t = 2c \\, e^t$, so $c = 1/2$ and $y_p = \\frac{1}{2} t e^t$.\n\nNon-example: $f(t) = 1/t$ on $(0, \\infty)$ is not a finite sum of the terms above, so it has no annihilator.\nSo the method does not apply to $y'' - y = 1/t$, and [[variation of parameters|0Y6TH5PQVS]] is used instead.\n",
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