Question

Are the following differential equations linear? Explain your reasoning. $$ y^{\prime}=x^{3}+e^{x} $$

Short answer

Yes, it is a linear differential equation; it fits the standard linear form.

Step-by-step solution

Identify the Order of the Differential Equation

The given differential equation is \( y' = x^3 + e^x \). The prime symbol (\('\)) indicates that this is a first-order differential equation. First-order differential equations involve the first derivative \( y' \).

Check for Linearity Conditions

A differential equation is linear if it can be expressed in the form \( a_0(x) y + a_1(x) y' = g(x) \), where \( a_0(x) \), \( a_1(x) \), and \( g(x) \) are functions of \( x \) only, and \( y \) and its derivatives appear to the power of 1.

Analyze the Given Equation

Rewrite the given equation \( y' = x^3 + e^x \) in the form \( 0 \, y + 1 \cdot y' = x^3 + e^x \). Here, \( a_0(x) = 0 \), \( a_1(x) = 1 \), and \( g(x) = x^3 + e^x \). This fits the standard linear form.

Confirm Linearity

In the given equation, \( y \) or its derivative \( y' \) is not raised to any power other than 1, nor is it multiplied by another \( y \) or \( y' \). Therefore, the differential equation satisfies the condition of linearity.

Key concepts

First-Order Differential Equations

A first-order differential equation involves only the first derivative of a function. In the equation given, \( y' = x^3 + e^x \), the notation \( y' \) indicates the derivative of \( y \) with respect to \( x \). This tells us that we are dealing with a first-order differential equation because it only involves \( y' \) which is the first derivative.First-order differential equations are commonly found in modeling natural phenomena like growth rates, cooling processes, and electrical circuits. Understanding them is crucial as they often form the foundation for more complex equations later on.Key characteristics of first-order differential equations include:

• They involve the function \( y \) and its first derivative \( y' \).
• They do not involve higher derivatives like \( y'' \) (second derivative with respect to \( x \)).

Being able to identify the order of a differential equation is a necessary skill. This step helps assess the complexity and methods available for solving it.

Linearity Conditions

To determine the linearity of a differential equation, it must fit within a specific form. A first-order differential equation is linear if it can be expressed as:\[ a_0(x) y + a_1(x) y' = g(x) \]Here, \( a_0(x) \) and \( a_1(x) \) are functions depending only on \( x \), and \( g(x) \) is another function of \( x \). Importantly, in a linear differential equation, none of the terms involving \( y \) or its derivatives are raised to any power other than 1.Let's rewrite our given equation \( y' = x^3 + e^x \) according to the linear form:- \( a_0(x) = 0 \)- \( a_1(x) = 1 \)- \( g(x) = x^3 + e^x \)By comparing, we see the differential equation adheres to the linearity conditions because the powers of \( y \) and \( y' \) meet the requirement. Only if these conditions are satisfied can the equation be considered linear.

Differential Equations Analysis

Performing differential equations analysis involves understanding the structure and properties of the equation. It enables us to decide the method for finding solutions.The given equation \( y' = x^3 + e^x \) fits the linear form, which simplifies the solution process because linear first-order differential equations typically have well-known solution techniques. Analyzing such equations generally involves:

• Determining if a solution exists and is unique, via conditions like linearity.
• Applying specific solution methods, such as the integrating factor method or separation of variables when applicable.

This type of analysis is not only about solving the equation but understanding its behavior. It involves checking whether the equation describes a realistic scenario based on initial values and constraints.Thus, differential equations analysis plays a vital role in making predictions and modeling scenarios across different domains like physics, engineering, and economics.