---
tag: 794N251WY8
kind: concept
name: Annihilator method
msc: [34A30, 44A45]
sources:
  - url: https://en.wikipedia.org/wiki/Annihilator_method
same_as:
  wikidata: Q4769489
  wikipedia: Annihilator method
---

## Summary

The annihilator method finds a particular solution of a [[linear differential equation|0QQRP17Z7S]] $L y = f$ with constant coefficients.
It applies when the right side $f$ itself solves some homogeneous constant-coefficient equation $A f = 0$.
Applying $A$ turns the problem into the homogeneous equation $A L y = 0$, whose solutions are known, and this tells exactly which trial functions to use.
It is the systematic form of the [[method of undetermined coefficients|0XRB469Z32]].

## Explanations

### L1 human

Some right-hand sides, such as $e^{2t}$, $t^3$ or $\cos t$, can be wiped out by a simple differential operator: $(D - 2) e^{2t} = 0$, $D^4 t^3 = 0$, $(D^2 + 1)\cos t = 0$.
Applying this wiping operator to both sides of the equation leaves an equation with right side zero.
Its solutions show the exact shape of a particular solution, and only some constants remain to be found.