Problem 23 - A First Course in Differential Equations with Modeling Applications

Question

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

Step-by-Step Solution

Answer

The operator that annihilates the function is \((D + 1)(D - 1)^3\).

Step 1: Identify Individual Functions

The given function is a combination of three different functions: \( f_1(x) = e^{-x} \), \( f_2(x) = 2xe^x \), and \( f_3(x) = -x^2 e^x \). We need to find the differential operator for each of these individually.

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

The function \(e^{-x}\) is a solution to the differential equation \((D + 1) y = 0\). Thus, the annihilator for \(e^{-x}\) is \((D + 1)\).

Step 3: Annihilator for \(2xe^x\)

The function \(xe^x\) is a solution to the differential equation \((D - 1)^2 y = 0\). Therefore, the annihilator for \(2xe^x\) is \((D - 1)^2\).

Step 4: Annihilator for \(-x^2 e^x\)

The function \(x^2 e^x\) is a solution to the differential equation \((D - 1)^3 y = 0\). So, the annihilator for \(-x^2 e^x\) is \((D - 1)^3\).

Step 5: Combine Annihilators

The overall differential operator that annihilates the entire function \(e^{-x} + 2xe^x - x^2e^x\) should be able to annihilate each component. So, the combined operator is the product: \((D + 1)(D - 1)^3\).

Key Concepts

Annihilator

An annihilator in the context of differential equations is a special type of differential operator. It is used to 'wipe out' or 'annihilate' a given function. This means that when applied to the function, it yields zero. To find the annihilator of a function, you must first identify its form. For example:

• The function \(e^{-x}\) is annihilated by the operator \((D + 1)\).
• The function \(xe^x\) is annihilated by \((D - 1)^2\).
• The function \(x^2 e^x\) is annihilated by \((D - 1)^3\).

These operators are derived from the characteristic equation of the function, linking directly to its differentiation properties.Once you have the annihilators of individual terms, you can combine them to create a comprehensive operator. This combined operator, when applied, will make the entire function equal to zero.

Linear Differential Equations

Linear differential equations involve derivatives of a function or functions. They have the property of linearity, meaning they can be expressed as a linear combination of the functions and their derivatives. This property makes them particularly suitable for solving equations involving unknown functions.A linear differential equation takes the general form: \[ L(y) = f(x) \]where \(L\) is a linear differential operator, \(y\) is the unknown function, and \(f(x)\) is a given function.These equations are crucial in fields like engineering, physics, and mathematics for modeling various phenomena such as motion, heat, and sound.The process to solve them often involves using an annihilator to find the unknown functions, breaking down complex problems into manageable parts.

Function Decomposition

Function decomposition is a method used to simplify complex functions by breaking them down into simpler parts. This is especially useful in the context of differential equations. The given problem asks to identify the annihilators for a combination of functions like \(e^{-x}\), \(2xe^x\), and \(-x^2 e^x\). These are individual components that together form a complex expression. By decomposing the function into these simpler parts, we can handle each one separately. This allows for the determination of an annihilator for each part. The sum of these simpler functions allows you to form a clear understanding of how the comprehensive function operates and simplifies the task of finding the overall operator. The combined use of these simpler operators helps solve complex equations efficiently.Decomposition is key, as it allows for a more systematic approach to tackling differential equations, making the solving process much more feasible.