Find A Differential Operator That Annihilates The Given Function

Finding a Differential Operator that Annihilates a Given Function

Finding a differential operator that annihilates a given function is a crucial technique in solving linear differential equations, particularly those with constant coefficients. This process involves identifying a differential operator, often represented as a polynomial in the derivative operator D (where D = d/dx), that, when applied to the function, results in zero. This article will explore this technique in detail, covering various function types and providing a step-by-step approach to finding the annihilating operator. Understanding this concept is key to mastering techniques like the method of undetermined coefficients and variation of parameters.

Introduction: Understanding Annihilation

A differential operator, denoted by L, is an expression involving derivatives. For example, L could be D² + 3D + 2, meaning L[y] = y'' + 3y' + 2y, where y is a function of x. An operator annihilates a function if, when applied to the function, the result is zero. In other words, L[y] = 0. Our goal is to find the operator L that makes this equation true for a specific function y.

Finding Annihilating Operators for Common Functions

The approach to finding an annihilating operator depends heavily on the form of the given function. Here's a breakdown for several common function types:

1. Polynomials:

If the function is a polynomial of degree n, say y(x) = a₀ + a₁x + a₂x² + ... + aₙxⁿ, the annihilating operator is simply Dⁿ⁺¹. This is because each successive differentiation reduces the degree of the polynomial until, after n+1 differentiations, we are left with zero.

Example: Let y(x) = 3x² - 2x + 1. The degree of the polynomial is 2, so the annihilating operator is D³ (the third derivative). Applying D³ to y(x) gives:

D[y(x)] = 6x - 2 D²[y(x)] = 6 D³[y(x)] = 0

2. Exponential Functions:

For functions of the form y(x) = e^(ax), the annihilating operator is (D - a). This is easily verified:

(D - a)[e^(ax)] = D[e^(ax)] - ae^(ax) = ae^(ax) - a*e^(ax) = 0

Example: If y(x) = 5e^(2x), the annihilating operator is (D - 2).

For a sum of exponential functions, the annihilating operator is the product of the individual annihilating operators.

Example: If y(x) = 2e^(3x) + 7e^(-x), the annihilating operator is (D - 3)(D + 1).

3. Trigonometric Functions:

Trigonometric functions require a slightly more involved approach. For functions of the form y(x) = cos(bx) or y(x) = sin(bx), the annihilating operator is D² + b². This stems from the second-order derivatives of sine and cosine.

Example: If y(x) = 3cos(4x), the annihilating operator is D² + 16.

To verify:

D[3cos(4x)] = -12sin(4x) D²[3cos(4x)] = -48cos(4x) (D² + 16)[3cos(4x)] = -48cos(4x) + 16(3cos(4x)) = 0

Similarly, for functions involving both sine and cosine with the same frequency, the same annihilating operator applies. Linear combinations of sine and cosine with the same argument are annihilated by the same operator.

4. Products of Exponential and Trigonometric Functions:

Functions of the form y(x) = e^(ax)cos(bx) or y(x) = e^(ax)sin(bx) require a combination of the techniques above. The annihilating operator is (D² - 2aD + a² + b²)

Example: If y(x) = e^(2x)sin(3x), the annihilating operator is (D² - 4D + 13).

5. Polynomials multiplied by Exponential Functions:

For functions of the form y(x) = P(x)e^(ax), where P(x) is a polynomial of degree n, the annihilating operator is (D - a)^(n+1).

Example: If y(x) = (x² + 2x)e^(3x), the annihilating operator is (D - 3)³.

Step-by-Step Guide to Finding Annihilating Operators

Let's outline a structured approach:

Identify the Function Type: Determine the primary components of the function: polynomial, exponential, trigonometric, or combinations thereof.
Apply Appropriate Annihilating Operators: Based on the function type, select the corresponding annihilating operator from the rules described above. If the function is a sum of terms, use the product of the annihilating operators for each term.
Verify (Optional): To confirm, apply the obtained operator to the original function. The result should be zero.

Advanced Cases and Linear Combinations

When dealing with linear combinations of different function types, the approach becomes more sophisticated. The annihilating operator for a linear combination is the least common multiple (LCM) of the individual annihilating operators. Finding the LCM of differential operators often involves factoring the operators into their linear factors. This can be computationally intensive for high-order operators. However, it's fundamentally based on finding the individual annihilators and then building the combined operator.

Applications in Solving Differential Equations

The concept of annihilating operators finds its prime application in solving non-homogeneous linear differential equations with constant coefficients. The method of undetermined coefficients relies heavily on this technique. Once the annihilating operator for the non-homogeneous term is found, it's applied to the entire differential equation, transforming it into a homogeneous equation that's easier to solve.

Frequently Asked Questions (FAQ)

Q1: What if I have a function that is not in the standard forms discussed above?

A1: For more complex functions, you might need to use a combination of techniques or decompose the function into simpler parts for which you can find annihilating operators. In some cases, no simple annihilating operator exists.

Q2: Can the annihilating operator be non-unique?

A2: Yes, there can be multiple annihilating operators for a given function. For instance, D² and D³ both annihilate any linear function. However, the minimal annihilating operator – the one with the lowest order – is usually the most useful.

Q3: How does this relate to the method of undetermined coefficients?

A3: In the method of undetermined coefficients, you guess a particular solution to a non-homogeneous differential equation based on the form of the forcing function (the non-homogeneous term). The form of this guess is directly informed by the annihilating operator of the forcing function.

Q4: What are the limitations of this method?

A4: The method is primarily suited for linear differential equations with constant coefficients. For equations with variable coefficients or nonlinear equations, different techniques are required. Also, for very complicated functions, finding the annihilator can become computationally challenging.

Conclusion

Finding a differential operator that annihilates a given function is a powerful technique with significant implications in the field of differential equations. By understanding the annihilating operators for fundamental function types and their combinations, you can effectively solve a broad range of non-homogeneous linear differential equations using methods like undetermined coefficients. This process, though mathematically rigorous, is inherently systematic and learnable with practice. Mastering this technique forms a solid foundation for more advanced studies in differential equations and related fields.