In the context of differential equations, an annihilator is a differential operator that, when applied to a function, yields zero. This concept is particularly useful when solving nonhomogeneous linear differential equations using the method of undetermined coefficients or annihilator method.
If f(x)f(x) is a known function and there exists a differential operator LL such that L[f(x)]=0L[f(x)] = 0, then LL is called an annihilator of f(x)f(x). The goal is to identify such an operator, often denoted by DD (where D=ddxD = \frac{d}{dx}).
Common Annihilators and How to Identify Them
To find a differential operator that annihilates a function, you must recognize the type of function you’re working with—polynomial, exponential, trigonometric, or a product—and match it to its known annihilator.
Annihilators for Basic Functions
Polynomial xnx^n: (Dn+1)(D^{n+1})
Exponential eaxe^{ax}: (D−a)(D – a)
Sine or Cosine sin(bx),cos(bx)\sin(bx), \cos(bx): (D2+b2)(D^2 + b^2)
Annihilators for Products
For example, if the function is xneaxx^n e^{ax}, the annihilator becomes (D−a)n+1(D – a)^{n+1}. This is due to the combination of polynomial and exponential behavior.
Annihilators for Linear Combinations
If you have a linear combination of functions (e.g., f(x)=e2x+sin(x)f(x) = e^{2x} + \sin(x)), the annihilator will be the least common multiple (LCM) of the individual annihilators:
Annihilator of e2x=(D−2)\text{Annihilator of } e^{2x} = (D – 2)
Annihilator of sin(x)=(D2+1)\text{Annihilator of } \sin(x) = (D^2 + 1)
Combined Annihilator: (D−2)(D2+1)(D – 2)(D^2 + 1)
Step-by-Step: Finding an Annihilator for a Given Function
To understand how to apply annihilators, let’s go through a structured process.
Step 1: Analyze the Function Type
First, categorize your function. Is it:
A polynomial?
An exponential?
A sine or cosine?
A product of two types?
For example, if your function is f(x)=x2e3xf(x) = x^2e^{3x}, then it is a product of a polynomial and an exponential.
Step 2: Match to Known Annihilator
Use standard annihilator tables:
eaxe^{ax}: D−aD – a
xnx^n: Dn+1D^{n+1}
So for x2e3xx^2e^{3x}, the annihilator is (D−3)3(D – 3)^3.
Step 3: Apply the Operator to Verify
Check your result:
(D−3)3[x2e3x]=0(D – 3)^3 [x^2e^{3x}] = 0
If this simplifies to zero, your annihilator is correct.
Examples and Applications of Annihilators
Using annihilators becomes extremely helpful when solving nonhomogeneous linear differential equations using operator methods.
Example 1: Exponential Function
Find the annihilator of f(x)=e2xf(x) = e^{2x}
Since e2xe^{2x} is exponential:
L=D−2L = D – 2
Check: (D−2)[e2x]=D[e2x]−2e2x=2e2x−2e2x=0(D – 2)[e^{2x}] = D[e^{2x}] – 2e^{2x} = 2e^{2x} – 2e^{2x} = 0
Example 2: Combination Function
Find the annihilator of f(x)=e2x+cos(3x)f(x) = e^{2x} + \cos(3x)
Annihilator of e2xe^{2x}: D−2D – 2
Annihilator of cos(3x)\cos(3x): D2+9D^2 + 9
Combined Annihilator: (D−2)(D2+9)(D – 2)(D^2 + 9)
Example 3: Product Function
Find the annihilator of f(x)=x2sin(4x)f(x) = x^2\sin(4x)
For x2x^2: D3D^3
For sin(4x)\sin(4x): D2+16D^2 + 16
Combined Annihilator: (D2+16)3(D^2 + 16)^3
When and Why to Use the Annihilator Method
The annihilator method is primarily used in solving nonhomogeneous linear differential equations where the right-hand side is a function like eaxe^{ax}, sin(bx)\sin(bx), or a combination. Instead of guessing a solution, you use differential operators to turn the original equation into a homogeneous equation of higher order.
Benefits of Using Annihilators
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Systematic approach – less trial and error
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Structured for linear operators – fits well into operator theory
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Works well with constant coefficient equations
Limitations of the Method
Not ideal for variable coefficient differential equations
Can become complex with multiple functions or high degrees
Doesn’t work easily with nonlinear equations
Conclusion: Mastering Annihilators for Advanced Problem Solving
Understanding how to find and use differential operators that annihilate functions can greatly simplify the process of solving linear differential equations. Once you know the annihilators for standard functions, you can quickly transform equations into solvable forms and reduce guesswork. Whether you’re a student, teacher, or engineer, mastering this tool will deepen your grasp of differential equations and operator methods.
