Find A Differential Operator That Annihilates The Given Function

Let's explore how to find a differential operator that annihilates a given function. This involves understanding the relationship between functions and differential operators, and applying techniques to construct operators that, when applied to the function, result in zero. The core concept revolves around identifying the fundamental solutions to homogeneous linear differential equations.

Introduction to Annihilators

An annihilator, in the context of differential equations, is a differential operator that, when applied to a given function, yields zero. More formally, if L is a differential operator and f(x) is a function, then L is an annihilator of f(x) if L(f(x)) = 0. The concept of annihilators is particularly useful in solving nonhomogeneous linear differential equations. By understanding annihilators, we can transform nonhomogeneous equations into homogeneous ones, making them easier to solve.

Think of it this way: A differential operator acts on a function, performing operations like differentiation and multiplication by constants. The annihilator is a specific operator tailored to "cancel out" the function, leaving nothing behind (zero).

Fundamental Concepts: Linear Differential Operators

Before diving into finding annihilators, it's crucial to understand linear differential operators. A linear differential operator L is an operator of the form:

L = a_n(x)Dⁿ + a_n-1(x)D^n-1 + ... + a₁(x)D + a₀(x)

where:

• D is the differential operator, D = d/dx. So, D(y) is dy/dx, D²(y) is d²y/dx², and so on.
• a_n(x), a_n-1(x), ..., a₁(x), a₀(x) are functions of x (often constants).

For example, L = D² + 3D + 2 is a linear differential operator. Applying it to a function y(x) gives:

L(y) = D²(y) + 3D(y) + 2y = d²y/dx² + 3dy/dx + 2y

A key property of linear differential operators is linearity:

• L(c₁y₁ + c₂y₂) = c₁L(y₁) + c₂L(y₂), where c₁ and c₂ are constants, and y₁ and y₂ are functions.

This property is fundamental to how annihilators work.

Identifying Basic Annihilators

Let's look at some basic functions and their corresponding annihilators. This provides a foundation for handling more complex functions.

• Function: e^ax

  • Annihilator: (D - a)

  • Explanation: The derivative of e^ax is ae^ax. Therefore, (D - a)(e^ax) = De^ax - ae^ax = ae^ax - ae^ax = 0.

• Function: xⁿ (where n is a non-negative integer)

  • Annihilator: Dⁿ⁺¹

  • Explanation: The n+1th derivative of xⁿ is zero. For example, if f(x) = x², then D(x²) = 2x, D²(x²) = 2, and D³(x²) = 0.

• Function: sin(bx) or cos(bx)

  • Annihilator: (D² + b²)

  • Explanation: Let's consider sin(bx). D(sin(bx)) = bcos(bx), and D²(sin(bx)) = -b²sin(bx). Therefore, (D² + b²)(sin(bx)) = -b²sin(bx) + b²sin(bx) = 0. A similar argument applies to cos(bx).

• Function: e^axsin(bx) or e^axcos(bx)

  • Annihilator: [(D - a)² + b²]

  • Explanation: This builds on the previous two. The term (D - a) handles the exponential part, and the b² handles the sinusoidal part. Applying this operator involves a bit more algebra, but the result will be zero.

Finding Annihilators for More Complex Functions

Many functions are combinations of the basic types discussed above. The key to finding their annihilators is to break them down into their constituent parts and combine the corresponding annihilators.

Rule: If a function f(x) is a sum of functions f₁(x), f₂(x), ..., f_n(x), and L_i is an annihilator for f_i(x), then the product L₁L₂...L_n is an annihilator for f(x). In other words, apply the annihilators sequentially.

Example 1: Find an annihilator for f(x) = 5e^2x - 3x².

1. Identify the constituent functions: 5e^2x and -3x².
2. Find the annihilator for e^2x: (D - 2). Since the constant multiple doesn't affect the annihilator (because of linearity), (D - 2) also annihilates 5e^2x.
3. Find the annihilator for x²: D³. This also annihilates -3x².
4. Combine the annihilators: D³(D - 2).

Therefore, D³(D - 2) is an annihilator for 5e^2x - 3x². Note that the order of application matters since differential operators don't always commute. In this case, D³(D-2) is the correct form; applying (D-2)D³ would also annihilate the function, but convention usually favors placing the D operator with the highest power first.

Example 2: Find an annihilator for f(x) = xcos(3x).

1. Recognize the form: This is a product of x and cos(3x). The product rule complicates things slightly. We need to consider what happens when we repeatedly apply the basic annihilators.

2. Annihilator for cos(3x): (D² + 9).

3. Now consider what happens when we apply (D² + 9) to xcos(3x):

   (D² + 9)(xcos(3x)) = D²(xcos(3x)) + 9xcos(3x)

   Calculate the derivatives:

   • D(xcos(3x)) = cos(3x) - 3xsin(3x)
   • D²(xcos(3x)) = -3sin(3x) - 3sin(3x) - 9xcos(3x) = -6sin(3x) - 9xcos(3x)

   Therefore:

   (D² + 9)(xcos(3x)) = -6sin(3x) - 9xcos(3x) + 9xcos(3x) = -6sin(3x)

   Notice that we didn't get zero, but we did get a simpler function: -6sin(3x).

4. Now, find an annihilator for -6sin(3x). We already know it's (D² + 9).

5. Therefore, the annihilator for xcos(3x) is (D² + 9)(D² + 9) = (D² + 9)².

Example 3: Find an annihilator for f(x) = e^-xsin(2x) + 3x.

1. Constituent functions: e^-xsin(2x) and 3x.
2. Annihilator for e^-xsin(2x): [(D + 1)² + 4] = (D² + 2D + 5). Here, a = -1 and b = 2.
3. Annihilator for 3x: D².
4. Combined annihilator: D²(D² + 2D + 5).

General Rules and Guidelines

Here's a summary of rules and guidelines for finding annihilators:

• Linearity: If L₁ annihilates f₁(x) and L₂ annihilates f₂(x), then L₁L₂ annihilates c₁f₁(x) + c₂f₂(x), where c₁ and c₂ are constants.
• Polynomials: For xⁿ, the annihilator is Dⁿ⁺¹.
• Exponentials: For e^ax, the annihilator is (D - a).
• Sines and Cosines: For sin(bx) or cos(bx), the annihilator is (D² + b²).
• Exponentially Modulated Sines and Cosines: For e^axsin(bx) or e^axcos(bx), the annihilator is [(D - a)² + b²].
• Products: If a function involves a product (like xcos(bx) or xe^ax), you may need to apply the basic annihilator multiple times to completely eliminate the function. The annihilator for xⁿe^ax is (D-a)ⁿ⁺¹. The annihilator for xⁿcos(bx) or xⁿsin(bx) is (D² + b²)ⁿ⁺¹. The annihilator for xⁿe^axcos(bx) or xⁿe^axsin(bx) is [(D - a)² + b²]ⁿ⁺¹.
• Sums: For a sum of functions, find the annihilator for each individual function and then multiply them together.

Applications of Annihilators in Solving Differential Equations

Annihilators are a powerful tool for solving nonhomogeneous linear differential equations of the form:

a_n(x)y⁽ⁿ⁾ + a_n-1(x)y^(n-1) + ... + a₁(x)y' + a₀(x)y = g(x)

where g(x) is a non-zero function. The method of annihilators provides a systematic way to find a particular solution y_p to this equation.

Here's the general procedure:

1. Find the complementary solution y_c: Solve the associated homogeneous equation:

   a_n(x)y⁽ⁿ⁾ + a_n-1(x)y^(n-1) + ... + a₁(x)y' + a₀(x)y = 0

   This involves finding the roots of the characteristic equation and constructing the general solution y_c based on those roots.

2. Find an annihilator L for g(x): This is the crucial step we've been discussing. Determine the appropriate differential operator L that annihilates the nonhomogeneous term g(x).

3. Apply the annihilator to the original differential equation: Apply the operator L to both sides of the original nonhomogeneous equation:

   L[a_n(x)y⁽ⁿ⁾ + a_n-1(x)y^(n-1) + ... + a₁(x)y' + a₀(x)y] = L[g(x)]

   Since L annihilates g(x), the right side becomes zero:

   L[a_n(x)y⁽ⁿ⁾ + a_n-1(x)y^(n-1) + ... + a₁(x)y' + a₀(x)y] = 0

   This transforms the nonhomogeneous equation into a homogeneous equation of higher order.

4. Solve the new homogeneous equation: Solve the new homogeneous equation obtained in step 3. The general solution to this equation will contain both the complementary solution y_c (which you already found) and a particular solution y_p.

5. Identify the form of y_p: The particular solution y_p will consist of terms that are not present in the complementary solution y_c. This is because any terms present in y_c will already satisfy the homogeneous part of the original equation.

6. Substitute y_p into the original nonhomogeneous equation and solve for the coefficients: Substitute the assumed form of y_p into the original nonhomogeneous equation. This will give you a set of algebraic equations that you can solve for the unknown coefficients in y_p.

7. Write the general solution: The general solution to the nonhomogeneous equation is the sum of the complementary solution and the particular solution:

   y = y_c + y_p

Example: Solve the differential equation y'' - y = x using the method of annihilators.

1. Complementary solution: The characteristic equation is r² - 1 = 0, so r = ±1. Therefore, y_c = c₁e^x + c₂e^-x.

2. Annihilator for x: The annihilator for x is D².

3. Apply the annihilator: Applying D² to both sides of the original equation gives:

   D²(y'' - y) = D²(x)

   y⁽⁴⁾ - y'' = 0

4. Solve the new homogeneous equation: The characteristic equation for this new equation is r⁴ - r² = 0, which factors as r²(r² - 1) = 0. The roots are r = 0, 0, 1, -1. The general solution is:

   y = c₁e^x + c₂e^-x + c₃ + c₄x

5. Identify the form of y_p: Comparing this to y_c = c₁e^x + c₂e^-x, we see that the terms c₁e^x and c₂e^-x are already in y_c. Therefore, the particular solution has the form:

   y_p = A + Bx

6. Substitute y_p into the original equation: Substitute y_p = A + Bx into y'' - y = x:

   (A + Bx)'' - (A + Bx) = x

   0 - A - Bx = x

   Equating coefficients, we get:

   • -A = 0 => A = 0
   • -B = 1 => B = -1

   Therefore, y_p = -x.

7. Write the general solution:

   y = y_c + y_p = c₁e^x + c₂e^-x - x

Common Mistakes and How to Avoid Them

• Incorrectly Identifying the Annihilator: The most common mistake is choosing the wrong annihilator for a given function. Double-check the basic annihilators and the rules for combining them.
• Forgetting to Apply the Annihilator to the Entire Equation: Make sure to apply the annihilator to both sides of the differential equation.
• Not Considering Terms Already in the Complementary Solution: When determining the form of y_p, remember to exclude any terms that are already present in the complementary solution y_c. Failing to do so will lead to a redundant solution and make it impossible to solve for the coefficients.
• Algebra Errors: Solving for the coefficients in y_p often involves a fair amount of algebra. Be careful to avoid arithmetic errors.
• Not Understanding the Underlying Principles: Rote memorization of annihilators is not enough. Understand why each annihilator works and how it relates to the derivatives of the function it annihilates.

Advanced Techniques and Considerations

• Repeated Roots: If the characteristic equation has repeated roots, the form of the particular solution needs to be adjusted accordingly. For example, if g(x) = x²e^ax and e^ax and xe^ax are solutions to the homogeneous equation, then y_p should include a term of the form Ax³e^ax + Bx⁴e^ax + Cx⁵e^ax.

• Using Software Packages: Software packages like Mathematica, Maple, and MATLAB can greatly simplify the process of finding annihilators and solving differential equations. These tools can handle complex calculations and symbolic manipulations that would be difficult or impossible to do by hand. However, it's still essential to understand the underlying principles so you can interpret the results correctly.

Conclusion

Finding a differential operator that annihilates a given function is a fundamental technique in solving nonhomogeneous linear differential equations. By understanding the relationship between functions and differential operators, applying the correct annihilators, and carefully following the steps outlined above, you can effectively solve a wide range of differential equations. Practice is key to mastering this technique. Work through numerous examples to solidify your understanding and build confidence in your ability to find annihilators and solve differential equations. The method of annihilators, when understood and applied correctly, provides a powerful and elegant approach to a significant class of problems in differential equations.