Find A Differential Operator That Annihilates The Given Function

Find a Differential Operator that Annihilates the Given Function

When working with linear differential equations, especially those with constant coefficients, it is often useful to identify a differential operator that sends a particular function to zero. Such an operator is called an annihilator because it “annihilates” the function, turning it into the trivial solution of a homogeneous equation. Knowing the annihilator simplifies the process of solving non‑homogeneous equations via the method of undetermined coefficients or the annihilator method. Below we explore the theory behind annihilators, outline a step‑by‑step procedure to construct them, and illustrate the technique with several common types of functions.

1. What Is an Annihilator?

In the context of differential equations, let

[ D = \frac{d}{dx} ]

denote the differentiation operator. Any polynomial in (D),

[ L(D) = a_n D^n + a_{n-1} D^{n-1} + \dots + a_1 D + a_0, ]

acts on a sufficiently differentiable function (f(x)) by repeated differentiation and linear combination.

We say that (L(D)) annihilates (f(x)) if

[L(D)[f(x)] = 0 \quad \text{for all } x \text{ in the domain}. ]

If such a polynomial exists, the set of all annihilators of (f(x)) forms an ideal in the ring of polynomials with constant coefficients; the minimal annihilator (the one of lowest degree) is particularly useful because it yields the simplest auxiliary equation.

2. General Strategy for Finding an Annihilator

The annihilator of a function depends on its functional form. The following steps work for any function that is a linear combination of terms of the types listed below (exponentials, polynomials, sines/cosines, and their products).

Identify the basic building blocks of the given function.
- Exponential: (e^{\alpha x})
- Polynomial: (x^m) - Sine/Cosine: (\sin(\beta x)) or (\cos(\beta x))
Determine the annihilator for each block using known formulas (see Table 1).
If the function is a product of blocks (e.g., (x^m e^{\alpha x}\sin(\beta x))), the annihilator is the least common multiple (LCM) of the individual annihilators, taking into account repeated roots.
Combine the annihilators for a sum of terms by taking the LCM of the annihilators of each summand.
Write the final operator as a polynomial in (D).

3. Annihilators of Elementary Functions

Function (f(x))	Minimal Annihilator (L(D))	Reasoning
(e^{\alpha x})	((D-\alpha))	(D[e^{\alpha x}] = \alpha e^{\alpha x}) → ((D-\alpha)[e^{\alpha x}] = 0)
(x^m) (polynomial of degree (m))	(D^{m+1})	Differentiating a polynomial (m+1) times yields zero
(\sin(\beta x)) or (\cos(\beta x))	((D^2+\beta^2))	Second derivative brings back the original function with a sign change
(x^m e^{\alpha x})	((D-\alpha)^{m+1})	Combine polynomial and exponential rules
(x^m \sin(\beta x)) or (x^m \cos(\beta x))	((D^2+\beta^2)^{m+1})	Polynomial factor raises the power of the quadratic annihilator
(e^{\alpha x}\sin(\beta x)) or (e^{\alpha x}\cos(\beta x))	((D-\alpha)^2+\beta^2) (i.e., (D^2-2\alpha D+(\alpha^2+\beta^2)))	Complex conjugate roots (\alpha \pm i\beta)
(x^m e^{\alpha x}\sin(\beta x)) or (x^m e^{\alpha x}\cos(\beta x))	(\big[(D-\alpha)^2+\beta^2\big]^{m+1})	Combine all three effects

Table 1 – Minimal annihilators for common building blocks.

4. Step‑by‑Step Examples

Example 1: Pure Exponential

Function: (f(x)=5e^{3x})

Building block: (e^{3x}) → annihilator ((D-3)).
Constant factor 5 does not affect the operator.
Annihilator: (L(D)=D-3).

Check: ((D-3)[5e^{3x}] = 5(3e^{3x})-5\cdot3e^{3x}=0).

Example 2: Polynomial Times Exponential

Function: (f(x)= (2x^2-7x+1)e^{-2x})

Exponential part (e^{-2x}) → ((D+2)).
Polynomial part is degree 2 → raise the exponential factor to power (m+1=3).
Annihilator: (L(D)=(D+2)^3).

Verification (sketch): Applying ((D+2)^3) differentiates three times, each time adding (2) times the function; the polynomial degree drops by one each differentiation, reaching zero after three steps.

Example 3: Trigonometric Function

Function: (f(x)=4\sin(5x)-3\cos(5x))

Both sine and cosine share the same frequency (\beta=5).
Annihilator for (\sin(5x)) or (\cos(5x)) is ((D^2+5^2) = (D^2+25)).
Since the function is a linear combination, the same operator works for the sum.
Annihilator: (L(D)=D^2+25).

Check: ((D^2+25)[\sin(5x)] = -25\sin(5x)+25\sin(5x)=0);

Example 4: Polynomial Times Trigonometric Function

Function: (f(x)=x^3\sin(2x))

Polynomial part (x^3) → annihilator ((D^2+2^2) = (D^2+4)).
Trigonometric part (\sin(2x)) → annihilator ((D^2+2^2) = (D^2+4)).
Combining the two, the annihilator is ((D^2+4)^{3+1} = (D^2+4)^4).

Verification: Applying ((D^2+4)^4) differentiates four times, each time adding (4) times the function. The polynomial degree drops by one each differentiation, reaching zero after four steps.

Example 5: Exponential Times Trigonometric Function

Function: (f(x)=2e^{x}\sin(3x))

Exponential part (e^{x}) → annihilator ((D-1)).
Trigonometric part (\sin(3x)) → annihilator ((D^2+3^2) = (D^2+9)).
Combining the two, the annihilator is ((D-1)(D^2+9) = D^3-D^2+9D-9).

Verification: Evaluating (f(x)) and applying this annihilator yields a zero function.

5. Conclusion

The minimal annihilator for a given function is a crucial concept in solving differential equations. By understanding the relationship between the function's components (polynomials, exponentials, trigonometric functions, and their combinations) and their corresponding annihilators, we can efficiently find solutions to linear differential equations. This table and the examples demonstrate a systematic approach to identifying these annihilators. While the calculations may seem complex at first, mastering this process unlocks a powerful tool for simplifying differential equation problems and finding their solutions. The ability to efficiently determine the minimal annihilator allows for a more streamlined approach to solving differential equations, ultimately leading to a deeper understanding of the underlying mathematical principles. The process is not always straightforward, especially with more intricate functions, but the principle of decomposition and the corresponding annihilator identification provide a solid foundation for tackling a wide range of differential equations.

The examples illustrate how to systematically determine the minimal annihilator for a variety of functions. For simple polynomials, exponentials, or trigonometric functions, the annihilator is straightforward. When functions are combined—such as a polynomial times a trigonometric function, or an exponential times a trigonometric function—the annihilator is constructed by combining the individual annihilators, often with increased multiplicity to account for the polynomial degree. Verifying the annihilator by applying it to the function ensures correctness. This methodical approach is invaluable in solving linear differential equations, as it allows one to convert a nonhomogeneous equation into a homogeneous one by multiplying by the appropriate annihilator. Mastery of this process not only simplifies calculations but also deepens understanding of the structure of differential equations and their solutions. With practice, identifying minimal annihilators becomes an efficient and reliable tool in the broader context of differential equation theory and applications.