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 article delves into the methodology, providing a comprehensive understanding for students and professionals alike. We will explore various scenarios, from simple polynomial functions to more complex exponential and trigonometric functions, and even combinations thereof. Understanding this concept is fundamental to tackling advanced topics in differential equations and their applications in various scientific fields.

Introduction: What is an Annihilator?

In the realm of differential equations, an annihilator is a differential operator that, when applied to a given function, results in zero. In simpler terms, it's an operator that "kills" the function. This operator is typically a polynomial in the differentiation operator D, where D represents d/dx (the derivative with respect to x). For example, if we have the function f(x) = x², a suitable annihilator would be (D³), as the third derivative of x² is zero. The focus of this article is on finding these annihilators for various types of functions.

Finding Annihilators for Basic Functions:

Let's begin with the fundamental building blocks:

1. Polynomials:

For a polynomial function of degree n, p(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀, the annihilator is simply Dⁿ⁺¹. This is because the (n+1)th derivative of any polynomial of degree n will always be zero.

• Example: If f(x) = 3x² - 2x + 1 (a polynomial of degree 2), the annihilator is D³. Applying D³ to f(x) yields: D(f(x)) = 6x - 2 D²(f(x)) = 6 D³(f(x)) = 0

2. Exponential Functions:

For an exponential function of the form f(x) = e^rx, where 'r' is a constant, the annihilator is (D - r). This is because:

D(e^rx) = re^rx (D - r)(e^rx) = re^rx - re^rx = 0

• Example: If f(x) = e^2x, the annihilator is (D - 2).

3. Trigonometric Functions:

Trigonometric functions require a slightly different approach. Consider:

• f(x) = sin(rx): The annihilator is (D² + r²). This is because: D(sin(rx)) = rcos(rx) D²(sin(rx)) = -r²sin(rx) (D² + r²)(sin(rx)) = -r²sin(rx) + r²sin(rx) = 0

• f(x) = cos(rx): Similarly, the annihilator is (D² + r²).

• Example: If f(x) = cos(3x), the annihilator is (D² + 9).

Annihilators for Combinations of Functions:

The power of annihilators truly shines when dealing with combinations of the basic functions we've discussed. The key principle here is that the annihilator for a sum of functions is the least common multiple (LCM) of the individual annihilators. This requires understanding how to find the LCM of differential operators.

1. Linear Combinations of Polynomials and Exponentials:

Let's consider a function like f(x) = x²e^3x. The individual annihilators are D³ (for x²) and (D - 3) (for e^3x). To find the annihilator for the combined function, we need the LCM of D³ and (D-3). Since these operators are relatively prime, their LCM is simply their product: (D-3)D³. Applying this to f(x) will result in zero.

2. Linear Combinations of Exponentials and Trigonometric Functions:

Consider f(x) = e^2xsin(3x). The annihilator for e^2x is (D-2), and the annihilator for sin(3x) is (D²+9). However, it's not simply the product of these two operators; we need to carefully consider the interaction between them. The annihilator for f(x) will be ((D-2)² + 9) or, expanding it, (D² - 4D + 13).

Proof: Let's apply the operator: Let y = e^2xsin(3x). Then Dy = 2e^2xsin(3x) + 3e^2xcos(3x) D²y = 4e^2xsin(3x) + 6e^2xcos(3x) + 6e^2xcos(3x) - 9e^2xsin(3x) = -5e^2xsin(3x) + 12e^2xcos(3x) D³y = -10e^2xsin(3x) + 24e^2xcos(3x) - 15e^2xcos(3x) - 36e^2xsin(3x) = -46e^2xsin(3x) + 9e^2xcos(3x) ...and so on. The application becomes more complicated but the principle remains. The operator (D² - 4D + 13) effectively annihilates the function. This is a result from applying the method of undetermined coefficients and solving for the operator. The specific techniques used here often involve complex numbers, which are beyond the scope of this introductory discussion.

Higher-Order Differential Operators and Repeated Roots:

Things become more intricate when dealing with repeated roots in the characteristic equation of a differential operator. For example, if we have a function involving e^rx and xe^rx (like in the case of a repeated root in a homogeneous linear differential equation), the annihilator isn't simply (D-r). Instead, it becomes (D-r)². For multiple repeated roots, the power on the (D-r) factor increases accordingly. For example, if we have e^rx, xe^rx, and x²e^rx the annihilator would be (D-r)³. This directly reflects how repeated roots are handled when solving homogeneous linear differential equations.

A Step-by-Step Approach:

1. Identify the basic functions: Break down the given function into its constituent parts (polynomials, exponentials, sines, cosines).

2. Determine individual annihilators: Find the annihilator for each basic function using the rules outlined above.

3. Find the least common multiple (LCM): Determine the LCM of the individual annihilators. This might involve factoring and combining differential operators. This step requires careful algebraic manipulation.

4. Apply the LCM annihilator: Apply the resulting LCM annihilator to the original function. The result should be zero.

Frequently Asked Questions (FAQ):

• Q: What if I have a product of functions that aren't easily separable?

  • A: In such cases, the annihilator determination may become significantly more complex and might involve techniques beyond the scope of a basic introduction. Advanced methods such as the application of generating functions or the use of Laplace transforms may be required.

• Q: Can I use partial fractions to simplify the process of finding the LCM?

  • A: While not directly applicable to differential operators in the same way as with algebraic fractions, understanding partial fraction decomposition enhances one's ability to identify and manipulate the factors of differential operators, thus facilitating the LCM determination.

• Q: How are annihilators related to solving differential equations?

  • A: Annihilators form a cornerstone in the solution process for non-homogeneous linear differential equations with constant coefficients. By annihilating the non-homogeneous term, we can reduce the equation to a homogeneous equation, which is readily solvable using characteristic equations and known techniques.

Conclusion:

Finding the annihilator of a given function is a powerful technique within the realm of differential equations. While straightforward for simple functions, the process requires careful attention to detail, algebraic manipulation, and a thorough understanding of the properties of differential operators when dealing with combinations of functions and repeated roots. Mastering this skill is vital for anyone seeking a comprehensive understanding of differential equations and their practical applications in various scientific and engineering disciplines. The principles and methods outlined in this article equip learners with a solid foundation to tackle a wide range of problems. Further exploration into advanced techniques and specialized scenarios will enhance this foundational understanding.