Problem 19 - A First Course in Differential Equations with Modeling Applications

Question

Find a linear differential operator that annihilates the given function. $$\cos 2 x$$

Step-by-Step Solution

Answer

The differential operator is \( D^2 + 4 \).

Step 1: Understand the problem

We need to find a linear differential operator that, when applied to the function \( \cos 2x \), results in zero. This means we are looking for a differential equation solution where \( \cos 2x \) is the function that gets 'annihilated.'

Step 2: Consider the properties of the given function

The function \( \cos 2x \) is a trigonometric function, which can be represented in terms of exponential functions: \( \cos 2x = \frac{e^{2ix} + e^{-2ix}}{2} \). This transformation suggests that we are dealing with imaginary exponents.

Step 3: Identify the characteristic polynomial

The characteristic polynomial related to trigonometric functions of the form \( \cos(kx) \) is \( D^2 + k^2 \), where \( D = \frac{d}{dx} \) and \( k \) is the frequency of the cosine function. For \( \cos 2x \), \( k = 2 \).

Step 4: Construct the differential operator

Using the characteristic polynomial identified, the corresponding linear differential operator will be \( D^2 + 4 \), as we substitute \( k = 2 \) into \( D^2 + k^2 \).

Step 5: Verify that the operator annihilates the function

Apply the operator \( D^2 + 4 \) to \( \cos 2x \): First, differentiate \( \cos 2x \) twice: \( \frac{d}{dx} \cos 2x = -2 \sin 2x \) and \( \frac{d^2}{dx^2} \cos 2x = -4 \cos 2x \). Then substitute in the operator: \( D^2 \cos 2x = -4\cos 2x \), so \( (D^2 + 4)\cos 2x = -4\cos 2x + 4\cos 2x = 0 \). This confirms the operator annihilates \( \cos 2x \).

Key Concepts

Annihilator Method

The annihilator method is a technique used in differential equations to find a linear differential operator that, when applied to a given function, results in zero. This means the operator "annihilates" the function. It is a powerful tool in solving non-homogeneous differential equations since it helps to construct operators that can systematically remove particular solutions.

When searching for an annihilator of a function, you essentially look for an operator that makes the function disappear, much like solving a mathematical puzzle. The annihilator method is particularly useful in handling functions that are solutions to a homogeneous linear differential equation. To find an appropriate operator, knowledge of the function's properties, such as its type or form, is crucial.

• Identify the function to be annihilated.
• Represent it in a form that shows all its mathematical properties (like exponential or trigonometric forms).
• Use these properties to determine the differential operator that can annihilate the function.

In our problem, the task is to find an operator for the function \( \cos 2x \), utilizing its transformation to exponential form to identify properties that guide the annihilation process.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in mathematics, especially in calculus and differential equations. These functions exhibit periodic behavior and are often defined in the context of oscillations and waves.

For example, the cosine function \( \cos x \) can be expressed in terms of exponential functions using Euler's formula, given by \( \cos 2x = \frac{e^{2ix} + e^{-2ix}}{2} \). This transformation exposes hidden properties of the function that make it easier to manipulate, particularly in differential equations. By transforming cosine into exponentials, complex number methods can be employed, simplifying operations that might otherwise seem cumbersome.

Trigonometric identities and transformations are essential tools when dealing with operators. In solving our exercise, recognizing \( \cos 2x \) in its exponential form enables us to identify the corresponding characteristic polynomial necessary for determining the differential operator.

Characteristic Polynomial

The characteristic polynomial is a critical component in finding differential operators for trigonometric functions. When dealing with functions like \( \cos(kx) \), their characteristic polynomial is typically of the form \( D^2 + k^2 \). This arises from the observation that trigonometric functions can be represented as exponential functions, leading to equations where derivatives of the form \( D = \frac{d}{dx} \) naturally follow.

In the case of our function \( \cos 2x \), the characteristic polynomial is \( D^2 + 2^2 \), leading directly to the operator \( D^2 + 4 \). This relationship is important because it identifies the operator needed to annihilate the function, turning it to zero when applied.

To construct the characteristic polynomial:

• Identify the form of the trigonometric function (determine \( k \)).
• Construct and simplify the polynomial \( D^2 + k^2 \).
• Verify by applying the operator to ensure it results in zero.

With the correct polynomial, like \( D^2 + 4 \) for our problem, these steps confirm the solution.