Problem 26 - A First Course in Differential Equations with Modeling Applications

Question

Find a linear differential operator that annihilates the given function. $$e^{-x} \sin x-e^{2 x} \cos x$$

Step-by-Step Solution

Answer

The operator \(((D + 1)^2 + 1)((D - 2)^2 + 1)\) annihilates the given function.

Step 1: Identify Functions and their Forms

The given function is \( e^{-x} \sin x - e^{2x} \cos x \). We recognize this as a linear combination of two separate functions: \( e^{-x} \sin x \) and \( e^{2x} \cos x \). Each of these can be addressed separately with different diffraction operator methods.

Step 2: Find the Annihilator for \( e^{-x} \sin x \)

To find the annihilating operator for a function of the form \( e^{ax} \sin(bx) \), use the operator \((D-a)^2 + b^2\). Here, \( a = -1 \) and \( b = 1 \), so the annihilator is \((D + 1)^2 + 1^2 = (D + 1)^2 + 1\).

Step 3: Find the Annihilator for \( e^{2x} \cos x \)

For a function \( e^{ax} \cos(bx) \), we again use \((D-a)^2 + b^2\). In this case, \( a = 2 \) and \( b = 1 \), thus the operator is \((D - 2)^2 + 1^2 = (D - 2)^2 + 1\).

Step 4: Combine the Operators

To annihilate the entire function, form the product of the individual annihilators from Steps 2 and 3: \(((D + 1)^2 + 1)((D - 2)^2 + 1)\). This combined operator will annihilate any linear combination of \( e^{-x} \sin x \) and \( e^{2x} \cos x \).

Key Concepts

Annihilator Method

The annihilator method is a powerful technique in solving nonhomogeneous differential equations. It involves finding a differential operator that, when applied to a given function, results in zero. This operator "annihilates" the function. This method simplifies many problems by transforming them into homogeneous differential equations, which are typically easier to solve.

• Identify the type of function needing annihilation.
• Select or derive the suitable annihilator for each function based on its form.
• Combine annihilators if dealing with multiple functions.

Understanding the form of the annihilator is crucial. For functions of the form \( e^{ax} \sin(bx) \) or \( e^{ax} \cos(bx) \), the annihilator is \( (D-a)^2 + b^2 \). This operator ensures that the given function accounts for both exponential and trigonometric parts, reducing it to zero when applied correctly.In the exercise, each component of the original function was separately annihilated before combining them. This demonstrates the annihilator method's principle of step-by-step reduction to achieve a single comprehensive operator. Working through this carefully will help grasp the ultimate goal: preparing the path toward solving more complex differential equations.

Differential Equations

Differential equations involve equations containing derivatives of a function. These equations describe various phenomena such as physical systems, growth models, and more. Learning how to solve differential equations is essential as they hint at understanding how different systems change over time.Two main types of differential equations exist:

• Ordinary Differential Equations (ODEs): Contain functions of one independent variable and their derivatives. The example in the exercise is an ordinary differential equation as it revolves around the function of a single variable \( x \).
• Partial Differential Equations (PDEs): Involve multiple independent variables and partial derivatives of the function. These are used in more complex scenarios like fluid dynamics or electromagnetic fields.

When solving differential equations, techniques like the annihilator method faciliate converting complex parts into simpler, universally addressable equations. This process is beneficial for both practical applications and theoretical understanding.

Function Annihilation

Function annihilation is a focused technique within the broader annihilator method. It involves deploying a specific operator to a function such that the operation results in zero. This process is crucial for reducing complicated differential equations into a simpler form or breaking them down into manageable parts.To apply function annihilation correctly:

• Recognize the structure of the given function.
• Select an appropriate differential operator that matches the function's pattern.
• Apply the operator to effectively "nullify" the function.

In the given exercise, focus was placed on annihilating each function component separately. For example, handling the \( e^{-x} \sin x \), the operator \((D + 1)^2 + 1\) was selected to match and annul the function. Similarly, for \( e^{2x} \cos x \), the operator \((D - 2)^2 + 1\) was used.Understanding function annihilation simplifies solving more extensive systems as it aids in isolating and addressing individual elements of equations. Mastery of this technique provides deeper insights into effective problem-solving strategies in mathematics.