Find A Differential Operator That Annihilates The Given Function

Finding a differential operator that annihilates a given function is a fundamental concept in differential equations, particularly useful in solving nonhomogeneous linear differential equations with constant coefficients. Annihilation, in this context, means that when the differential operator is applied to the function, the result is zero. This technique simplifies complex problems by transforming them into more manageable forms. This article will comprehensively cover how to find such differential operators, providing a step-by-step guide, theoretical explanations, and practical examples to ensure a thorough understanding.

Introduction

A differential operator is a combination of derivatives that, when applied to a function, yields another function. The goal is to find a specific operator that, when applied to a given function, results in zero. This operator is said to "annihilate" the function. The concept is primarily used to simplify the solution process for nonhomogeneous linear differential equations.

Basic Concepts

• Differential Operator: An expression involving derivatives, typically denoted as D, where D = d/dx. For instance, D² represents the second derivative d²/dx².
• Annihilator: A differential operator A(D) such that A(D)f(x) = 0, where f(x) is the function being annihilated.

Why Find Annihilators?

Annihilators are valuable because they allow us to transform a nonhomogeneous differential equation into a homogeneous one. Homogeneous equations are generally easier to solve. This transformation is a key step in the method of undetermined coefficients, a common technique for solving nonhomogeneous linear differential equations.

Steps to Find a Differential Operator That Annihilates a Function

Step 1: Identify the Function Type

The first step in finding an annihilator is to identify the type of function you are dealing with. Common types include:

• Polynomials
• Exponentials
• Sines and Cosines
• Combinations of these

Step 2: Determine the Appropriate Differential Operator

Each type of function requires a specific form of differential operator to annihilate it. Here are the common forms:

Polynomials

If f(x) is a polynomial of degree n, i.e., f(x) = a_0 + a_1x + a_2x² + ... + a_nx^n, the annihilator is D^(n+1).

• Example:
  • If f(x) = 5x² + 3x + 2, then n = 2, and the annihilator is D³.

Exponentials

If f(x) = e^(ax), the annihilator is (D - a).

• Example:
  • If f(x) = e^(3x), the annihilator is (D - 3).

Sines and Cosines

If f(x) = sin(bx) or f(x) = cos(bx), the annihilator is (D² + b²).

• Example:
  • If f(x) = sin(2x), the annihilator is (D² + 4).
  • If f(x) = cos(5x), the annihilator is (D² + 25).

Combinations

For combinations of functions, you may need to apply a sequence of annihilators.

• Example:
  • If f(x) = x*e^(2x), you first annihilate the exponential part with (D - 2). However, since there is an x term, you need to apply (D - 2)² to annihilate the entire function.

Step 3: Apply the Differential Operator

Once you have identified the appropriate differential operator, apply it to the function to verify that it results in zero. This step ensures you have chosen the correct operator.

Step 4: Combine Annihilators for Complex Functions

Sometimes, a function may consist of a sum of different types of terms. In such cases, you need to find the annihilator for each term and then combine them. This often involves finding the least common multiple of the individual annihilators.

Detailed Examples

Example 1: Polynomial Function

Let's find the annihilator for the function f(x) = 3x² - 2x + 5.

1. Identify the Function Type: This is a polynomial of degree 2.
2. Determine the Appropriate Differential Operator: The annihilator for a polynomial of degree n is D^(n+1). Here, n = 2, so the annihilator is D³.
3. Apply the Differential Operator:
   • D(3x² - 2x + 5) = 6x - 2
   • D²(3x² - 2x + 5) = D(6x - 2) = 6
   • D³(3x² - 2x + 5) = D(6) = 0 Thus, D³ annihilates f(x) = 3x² - 2x + 5.

Example 2: Exponential Function

Find the annihilator for the function f(x) = 4e^(5x).

1. Identify the Function Type: This is an exponential function.
2. Determine the Appropriate Differential Operator: The annihilator for e^(ax) is (D - a). Here, a = 5, so the annihilator is (D - 5).
3. Apply the Differential Operator:
   • (D - 5)(4e^(5x)) = D(4e^(5x)) - 5(4e^(5x)) = 20e^(5x) - 20e^(5x) = 0 Thus, (D - 5) annihilates f(x) = 4e^(5x).

Example 3: Sine Function

Find the annihilator for the function f(x) = 2sin(3x).

1. Identify the Function Type: This is a sine function.
2. Determine the Appropriate Differential Operator: The annihilator for sin(bx) is (D² + b²). Here, b = 3, so the annihilator is (D² + 9).
3. Apply the Differential Operator:
   • D(2sin(3x)) = 6cos(3x)
   • D²(2sin(3x)) = D(6cos(3x)) = -18sin(3x)
   • (D² + 9)(2sin(3x)) = -18sin(3x) + 9(2sin(3x)) = -18sin(3x) + 18sin(3x) = 0 Thus, (D² + 9) annihilates f(x) = 2sin(3x).

Example 4: Combination of Functions

Find the annihilator for the function f(x) = x*e^(2x).

1. Identify the Function Type: This is a combination of a polynomial and an exponential function.
2. Determine the Appropriate Differential Operator:
   • For e^(2x), the annihilator is (D - 2).
   • However, due to the presence of x, we need to apply the operator twice to account for the polynomial part. Thus, the annihilator is (D - 2)².
3. Apply the Differential Operator:
   • (D - 2)(x*e^(2x)) = D(x*e^(2x)) - 2(x*e^(2x)) = e^(2x) + 2x*e^(2x) - 2x*e^(2x) = e^(2x)
   • (D - 2)²(x*e^(2x)) = (D - 2)(e^(2x)) = 2e^(2x) - 2e^(2x) = 0 Thus, (D - 2)² annihilates f(x) = x*e^(2x).

Example 5: Sum of Functions

Find the annihilator for the function f(x) = 3x² + 2e^(-x) + sin(4x).

1. Identify the Function Type: This is a sum of a polynomial, an exponential, and a sine function.
2. Determine the Appropriate Differential Operator:
   • For 3x², the annihilator is D³.
   • For 2e^(-x), the annihilator is (D + 1).
   • For sin(4x), the annihilator is (D² + 16).
3. Combine the Annihilators:
   • The combined annihilator is D³(D + 1)(D² + 16).
4. Verify (Conceptual): Applying D³(D + 1)(D² + 16) to f(x) will indeed result in zero, as each component of the operator will annihilate its corresponding term in f(x).

Advanced Considerations

Repeated Roots

When dealing with functions like x^n e^(ax), the annihilator will involve repeated factors. For example, if f(x) = x²e^(3x), the annihilator is (D - 3)³. The power of (D - a) corresponds to n + 1, where n is the highest power of x.

Complex Roots

For functions involving sines and cosines, such as e^(ax)sin(bx) or e^(ax)cos(bx), the annihilator is [(D - a)² + b²].

• Example:
  • For f(x) = e^(2x)cos(3x), the annihilator is [(D - 2)² + 9] = (D² - 4D + 4 + 9) = (D² - 4D + 13).

Hyperbolic Functions

Hyperbolic functions like sinh(ax) and cosh(ax) can be expressed in terms of exponentials, so their annihilators can be found using similar techniques.

• sinh(ax) = (e^(ax) - e^(-ax))/2
• cosh(ax) = (e^(ax) + e^(-ax))/2 The annihilator for both sinh(ax) and cosh(ax) is (D² - a²).

Practical Applications

Solving Nonhomogeneous Differential Equations

The primary application of annihilators is in solving nonhomogeneous linear differential equations with constant coefficients. Consider the equation: y'' + 3y' + 2y = 4e^(x)

1. Find the Annihilator for the Nonhomogeneous Term:
   • The nonhomogeneous term is 4e^(x), and its annihilator is (D - 1).
2. Apply the Annihilator to the Entire Equation:
   • (D - 1)(y'' + 3y' + 2y) = (D - 1)(4e^(x))
   • (D - 1)(y'' + 3y' + 2y) = 0
3. Solve the Resulting Homogeneous Equation:
   • The new homogeneous equation is (D - 1)(D² + 3D + 2)y = 0, which simplifies to (D - 1)(D + 1)(D + 2)y = 0.
   • The general solution for this equation is y(x) = c_1e^(x) + c_2e^(-x) + c_3e^(-2x).
4. Determine the Form of the Particular Solution:
   • From the general solution, we identify the part that is not already present in the homogeneous solution of the original equation. In this case, c_1e^(x) is the term we are interested in.
   • The particular solution has the form y_p(x) = Axe^(x).
5. Find the Coefficients of the Particular Solution:
   • Substitute y_p(x) into the original nonhomogeneous equation and solve for A.
   • y_p'(x) = Ae^(x) + Axe^(x)
   • y_p''(x) = 2Ae^(x) + Axe^(x)
   • Substituting into the original equation:
     • (2Ae^(x) + Axe^(x)) + 3(Ae^(x) + Axe^(x)) + 2(Axe^(x)) = 4e^(x)
     • 5Ae^(x) + 6Axe^(x) = 4e^(x)
   • Comparing coefficients, we get 5A = 4, so A = 4/5.
6. Write the General Solution:
   • The general solution is y(x) = c_2e^(-x) + c_3e^(-2x) + (4/5)xe^(x).

Simplifying Differential Equations

By using annihilators, you can simplify the process of solving complex differential equations. It breaks down the problem into smaller, more manageable parts.

Common Mistakes to Avoid

1. Incorrectly Identifying the Function Type:
   • Misidentifying the function type can lead to using the wrong annihilator. Always double-check the form of the function before applying an operator.
2. Forgetting Repeated Roots:
   • When dealing with functions like x^n e^(ax), remember to account for the repeated roots by using (D - a)^(n+1).
3. Not Combining Annihilators Correctly:
   • When dealing with sums of different types of functions, make sure to find the correct combination of annihilators.
4. Algebraic Errors:
   • Be careful with algebraic manipulations when applying and simplifying differential operators.

Conclusion

Finding a differential operator that annihilates a given function is a powerful technique in the realm of differential equations. By understanding the different types of functions and their corresponding annihilators, one can simplify and solve complex nonhomogeneous linear differential equations. This article has provided a detailed, step-by-step guide with examples, advanced considerations, and practical applications to equip you with the knowledge to confidently tackle these problems. Remember to practice and apply these techniques to master the art of annihilation.