Question

Are the following differential equations linear? Explain your reasoning. $$ \frac{d y}{d x}=x^{2} y+\sin x $$

Short answer

Yes, the equation is linear because it matches the form \( \frac{d y}{d x} + P(x)y = Q(x) \).

Step-by-step solution

Identify the General Form of a Linear Differential Equation

A first-order linear differential equation can be written in the form \( \frac{d y}{d x} + P(x)y = Q(x) \), where \( P(x) \) and \( Q(x) \) are functions of \( x \) only.

Analyze the Given Differential Equation

The given differential equation is \( \frac{d y}{d x} = x^{2} y + \sin x \). In order to compare it to the form from Step 1, we need to rewrite it similarly if possible.

Reorganize and Compare

Reorganize the equation as \( \frac{d y}{d x} - x^{2} y = \sin x \). This matches the standard form \( \frac{d y}{d x} + P(x)y = Q(x) \) with \( P(x) = -x^2 \) and \( Q(x) = \sin x \).

Check Conditions for Linearity

In a linear differential equation, the term \( P(x)y \) should only have the function \( y \) to the first power, which is satisfied, as \( P(x) = -x^2 \) does not affect the linearity in terms of the functions themselves.

Key concepts

First-Order Differential Equations

First-order differential equations are a key concept in calculus that involve the rate of change of a variable with respect to another. When we say "first-order," we mean that the highest derivative in the equation is the first derivative. This simply means we're dealing with the first level of differentiation.

The general approach to solving these equations involves identifying its form. For instance, in the equation \( \frac{dy}{dx} = x^2y + \sin x \), the aim is to express it in a recognizable form that's easier to work with. This involves combining and reorganizing terms to make it fit a standard template that can be more easily interpreted and solved. In essence, these equations help us model dynamics where the change in one variable depends linearly or non-linearly on itself and possibly other variables.

Linearity Conditions

Linearity in differential equations relates to the nature of the terms involved. For an equation to be linear, it must adhere to a specific structure. This is typically expressed as \( \frac{d y}{d x} + P(x)y = Q(x) \), where \( P(x) \) and \( Q(x) \) are functions of \( x \).

• The function \( y \) should only appear to the first power.
• There shouldn't be any products, powers, or complex functions of \( y \) within terms.

In the given problem, you rearrange the equation to see if it fits into \( \frac{d y}{d x} + P(x)y = Q(x) \). If rearranged properly into this structure without any deviation from the linear form, the equation can be classified as linear.

Differential Equations Analysis

Analyzing differential equations involves pinpointing the structure of the equation. The purpose is to determine what type of equation you are dealing with and how to go about solving it.

To analyze effectively, you first need to compare the given equation to a known standard form. If it matches, you can confidently apply specific methods to solve it or assess its properties. In our example, the equation \( \frac{d y}{d x} = x^2 y + \sin x \) was reorganized to \( \frac{d y}{d x} - x^2y = \sin x \). This helps clearly identify \( P(x) \) as \( -x^2 \) and \( Q(x) \) as \( \sin x \), thus confirming it fits the linear structure.

Functions of x

In differential equations, understanding the role of functions of \( x \) is crucial. These functions, denoted as \( P(x) \) and \( Q(x) \) in our context, dictate the behavior and solution of the equation.

• \( P(x) \) modifies the relationship between \( dy/dx \) and \( y \), determining how \( y \) scales in relation to its derivative.
• \( Q(x) \) acts as an external input or driving force that can shift or influence the solution behavior independently of \( y \).

By examining these functions, we gain insights into the dynamics described by the equation. For the given equation, \( P(x) = -x^2 \) and \( Q(x) = \sin x \), they highlight the interaction between the rate of change and the solution function, playing a pivotal role in determining the equation's solution and behavior.