Annihilator method

Remarks

The 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 $t^k e^{\alpha t} \sin \beta t$. For instance, $t^k e^{\alpha t}$ is annihilated by $(D - \alpha)^{k+1}$, and $e^{\alpha t} \cos \beta t$ by $(D - \alpha)^2 + \beta^2$. Step 4 always has a solution, which justifies the trial functions of the method of undetermined coefficients.

Example: $y'' - y = e^t$. Here $L = D^2 - 1$ and $A = D - 1$, and $A L = (D - 1)^2 (D + 1)$ gives the solutions $e^t, t e^t, e^{-t}$. Removing $e^t$ and $e^{-t}$, which solve $L y = 0$, leaves $y_p = c\, t e^t$. Then $y_p'' - y_p = c (t + 2) e^t - c t e^t = 2c \, e^t$, so $c = 1/2$ and $y_p = \frac{1}{2} t e^t$.

Non-example: $f(t) = 1/t$ on $(0, \infty)$ is not a finite sum of the terms above, so it has no annihilator. So the method does not apply to $y'' - y = 1/t$, and variation of parameters is used instead.