Find A Differential Operator That Annihilates The Given Function

Finding a Differential Operator That Annihilates the Given Function

Differential operators that annihilate specific functions form a powerful tool in solving linear differential equations with constant coefficients. These operators, which are polynomials in the differentiation operator D, can "zero out" particular types of functions, simplifying the solution process for nonhomogeneous equations. Understanding how to construct these annihilators is essential for advanced applications in mathematics, physics, and engineering.

Understanding Differential Operators

A differential operator is a mathematical operator that involves differentiation. The most basic differential operator is D, defined as D = d/dx. Higher-order operators are formed by taking powers of D, such as D² = d²/dx², D³ = d³/dx³, and so on. These operators can be combined with constant coefficients to create polynomial expressions in D, such as (D² - 3D + 2) or (D³ + 4D).

When we apply a differential operator to a function, we perform the indicated differentiations. For example, applying the operator D² to the function f(x) = x³ gives: D²(x³) = D(3x²) = 6x.

The Concept of Annihilators

An annihilator of a function f(x) is a differential operator L(D) such that when applied to f(x), the result is zero: L(D)[f(x)] = 0.

This concept is particularly useful in solving nonhomogeneous linear differential equations. If we can find an annihilator for the nonhomogeneous term g(x), we can transform the original equation into a homogeneous one with a higher order, which is often easier to solve.

Finding Annihilators for Basic Functions

Different types of functions have characteristic annihilators:

1. Exponential functions: For f(x) = e^(ax), the annihilator is (D - a). Example: For e^(3x), (D - 3)[e^(3x)] = 3e^(3x) - 3e^(3x) = 0.

2. Polynomials: For a polynomial of degree n, the annihilator is D^(n+1). Example: For f(x) = x² (degree 2), D³[x²] = D²[2x] = D[2] = 0.

3. Sine and cosine functions: For f(x) = sin(bx) or cos(bx), the annihilator is (D² + b²). Example: For sin(2x), (D² + 4)[sin(2x)] = D²[sin(2x)] + 4sin(2x) = -4sin(2x) + 4sin(2x) = 0.

4. Exponential times polynomial: For f(x) = e^(ax)P(x) where P(x) is a polynomial of degree n, the annihilator is (D - a)^(n+1). Example: For xe^(2x), (D - 2)²[xe^(2x)] = (D - 2)[e^(2x)] = 0.

5. Exponential times sine/cosine: For f(x) = e^(ax)sin(bx) or e^(ax)cos(bx), the annihilator is [(D - a)² + b²]. Example: For e^x sin(3x), [(D - 1)² + 9][e^x sin(3x)] = 0.

Method for Finding Annihilators

To find a differential operator that annihilates a given function, follow these steps:

1. Identify the form of the function: Determine if the function is exponential, polynomial, trigonometric, or a combination of these.

2. Apply the appropriate annihilator pattern:

   • For e^(ax): use (D - a)
   • For x^n: use D^(n+1)
   • For sin(bx) or cos(bx): use (D² + b²)
   • For e^(ax)x^n: use (D - a)^(n+1)
   • For e^(ax)sin(bx) or e^(ax)cos(bx): use [(D - a)² + b²]

3. For sums of functions: Find individual annihilators for each term and multiply them together. Important: The annihilator for a sum is the product of the individual annihilators, provided the functions are linearly independent.

4. Verify the result: Apply the operator to the function to confirm it yields zero.

Examples of Finding Annihilators

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

• The term 5e^(2x) is annihilated by (D - 2).
• The term -3x² is a polynomial of degree 2, annihilated by D³.
• Since the terms are linearly independent, the annihilator is (D - 2)D³.

Verification: (D - 2)D³[5e^(2x) - 3x²] = (D - 2)D³[5e^(2x)] - (D - 2)D³[3x²] = (D - 2)[0] - (D - 2)[0] = 0

Example 2: Find an annihilator for f(x) = e^x sin(2x).

• This is an exponential times sine function with a = 1 and b = 2.
• The annihilator is [(D - 1)² + 4] = (D² - 2D + 5).

Verification: (D² - 2D + 5)[e^x sin(2x)] = e^x[(D² - 2D + 5)sin(2x)] (by exponential shift theorem) = e^x[-4sin(2x) - 4cos(2x) + 5sin(2x)] = e^x[sin(2x) - 4cos(2x)]? Wait, let's compute properly:

Actually, we should apply the operator directly: Let y = e^x sin(2x) y' = e^x sin(2x) + 2e^x cos(2x) = e^x(sin(2x) + 2cos(2x)) y'' = e^x(sin(2x) + 2cos(2x)) + e^x(2cos(2x) - 4sin(2x)) = e^x(-3sin(2x) + 4cos(2x))

Now (D² - 2D + 5)y = y'' - 2y' + 5y = e^x(-3sin(2x) + 4cos(2x)) - 2e^x(sin(2x) + 2cos(2x)) + 5e^x sin(2x) = e^x[(-3 - 2 + 5)sin(

Example 2 (continued)
To verify that ((D^{2}-2D+5)) indeed annihilates (y=e^{x}\sin(2x)), compute successive derivatives:

[ \begin{aligned} y &= e^{x}\sin(2x),\[4pt] y' &= e^{x}\sin(2x)+2e^{x}\cos(2x)=e^{x}\bigl(\sin(2x)+2\cos(2x)\bigr),\[4pt] y'' &= e^{x}\bigl(\sin(2x)+2\cos(2x)\bigr) +e^{x}\bigl(2\cos(2x)-4\sin(2x)\bigr)\ &=e^{x}\bigl(-3\sin(2x)+4\cos(2x)\bigr). \end{aligned} ]

Now apply the operator:

[ \begin{aligned} (D^{2}-2D+5)y &= y''-2y'+5y\ &=e^{x}\bigl(-3\sin(2x)+4\cos(2x)\bigr) -2e^{x}\bigl(\sin(2x)+2\cos(2x)\bigr) +5e^{x}\sin(2x)\ &=e^{x}\bigl[(-3-2+5)\sin(2x)+(4-4)\cos(2x)\bigr]=0. \end{aligned} ]

Thus ((D^{2}-2D+5)) is a valid annihilator.

3. Annihilators for Products of the Basic Forms

When a function is a product of two simpler pieces, the annihilator can often be obtained by combining the individual annihilators. The most common cases are:

Derivation sketch: The exponential factor merely shifts the differential operator, while the polynomial factor forces an extra power of the operator, and the trigonometric factor introduces the quadratic term. When both polynomial and trigonometric elements appear together, the quadratic factor is taken to a power equal to the polynomial degree + 1.

4. Solving Non‑Homogeneous Linear ODEs with Annihilators

Suppose we must solve

[ L[y]=g(x), ]

where (L) is a linear differential operator with constant coefficients and (g(x)) is a known forcing term. The annihilator method proceeds as follows:

1. Find an annihilator (A) for the non‑homogeneous term (g(x)).
   By the rules above, (A) is a product of simple factors matching the shape of (g(x)).

2. Apply (A) to both sides of the equation:
   [ A,L[y]=A,g(x)=0. ] The left‑hand side is now a homogeneous linear differential equation of higher order.

3. Solve the enlarged homogeneous equation.
   Its characteristic polynomial is the product of the characteristic polynomials of (L) and (A). Consequently the general solution consists of:

   • The complementary solution of the original homogeneous equation (L[y]=0);
   • The complementary solution of the enlarged equation that comes from the factors in (A).

4. Select the particular solution from the enlarged homogeneous solution that is not already part of the original complementary solution.
   If overlap occurs (i.e., a term of the guessed particular solution is already a solution of the homogeneous equation), multiply the guess by (x) enough times to remove the duplication (the “method of annihilators” equivalent of reduction of order).

5. Combine the complementary solution of the original problem with the particular solution to obtain the full solution.

Illustrative example
Solve

[ y''-3y'+2y = e^{x}\sin(2x). ]

• The homogeneous operator is (L=(D^{2}-3D+2)=(D-1)(D-2)); its complementary solution is (C_{1}e^{x}+C_{2}e^{2x}).

• (g(x)=e^{x}\sin(2x)) has annihilator (A=[(D-1)^{2}+4]=(D^{2}-2D+5)).

• Form the combined operator (A,L=(D^{2}-2D+5)(D-1)(D-2)).
  Its characteristic equation is ((r-1)(

r-2)(r^{2}-2r+5)=0), which yields roots (r=1, 2, 1+2i, 1-2i).

• The general solution to the enlarged homogeneous equation is (y_{h}=C_{1}e^{x}+C_{2}e^{2x}+C_{3}e^{x}\cos(2x)+C_{4}e^{x}\sin(2x)).

• Since (C_{1}e^{x}) and (C_{2}e^{2x}) are already part of the original complementary solution, we need a particular solution that isn't. We initially guess (y_{p}=e^{x}\sin(2x)), but this is already a solution to the enlarged homogeneous equation. Therefore, we multiply by (x) to obtain (y_{p}=x e^{x}\sin(2x)).

• The full solution is (y=C_{1}e^{x}+C_{2}e^{2x}+C_{3}e^{x}\cos(2x)+C_{4}e^{x}\sin(2x)+x e^{x}\sin(2x)).

5. Advantages and Limitations of the Annihilator Method

The annihilator method offers a powerful and systematic approach to solving non-homogeneous linear ODEs, particularly when the forcing function (g(x)) has a relatively simple form. Its strength lies in its ability to break down a complex problem into manageable steps, leveraging the properties of differential operators to simplify the equation. The method is particularly effective for forcing functions involving exponentials, polynomials, and trigonometric functions, as outlined in the table of annihilators.

However, the method isn't without its limitations.

• Complexity with Complex (g(x)): When (g(x)) is a complicated combination of functions, finding a suitable annihilator can be challenging or even impossible. The method relies on the existence of a relatively straightforward annihilator, and highly irregular or piecewise functions may not lend themselves to this approach.
• Redundancy and Overlap: The need to multiply by (x) repeatedly to avoid overlap between the particular solution and the complementary solution can become tedious and computationally intensive, especially for higher-order equations.
• Not a Universal Solution: The annihilator method is specifically tailored for linear ODEs with constant coefficients. It cannot be directly applied to equations with variable coefficients.
• Alternative Methods: For certain types of forcing functions, other methods like the method of variation of parameters might be more efficient or easier to implement.

Conclusion

The annihilator method provides a valuable tool in the arsenal of techniques for solving non-homogeneous linear ordinary differential equations with constant coefficients. By systematically applying annihilators to both the differential operator and the forcing function, the method transforms the original problem into a more manageable homogeneous equation. While it has limitations, particularly with complex forcing functions, its clarity and structured approach make it a preferred method for many common ODE problems. Understanding the underlying principles of annihilators—the way differential operators interact with various function types—not only facilitates the solution of ODEs but also deepens the appreciation for the mathematical structure governing these equations. The method’s elegance lies in its ability to harness the power of differential operators to simplify complex problems, offering a clear and efficient pathway to their solutions.