Question

Solve each differential equation. $$ \frac{d y}{d x}-\frac{y}{x}=x e^{x} $$

Short answer

The solution is \( y = xe^x + Cx \).

Step-by-step solution

Identify the Type of Differential Equation

The given differential equation \( \frac{dy}{dx} - \frac{y}{x} = xe^x \) is a first-order linear differential equation. It is in the standard form \( \frac{dy}{dx} + P(x)y = Q(x) \), with \( P(x) = -\frac{1}{x} \) and \( Q(x) = xe^x \).

Calculate Integrating Factor

The integrating factor (IF) is found using the formula \( IF = e^{\int P(x) \, dx} \). So, first calculate \( \int -\frac{1}{x} \, dx = -\ln|x| \). The integrating factor is \( e^{-\ln|x|} = \frac{1}{x} \).

Multiply Differential Equation by Integrating Factor

Multiply the entire differential equation \( \frac{dy}{dx} - \frac{y}{x} = xe^x \) by \( \frac{1}{x} \) to obtain \( \frac{1}{x} \frac{dy}{dx} - \frac{y}{x^2} = e^x \). This simplifies the equation into the form \( \frac{d}{dx}(\frac{y}{x}) = e^x \).

Integrate Both Sides

Integrate both sides of the equation \( \frac{d}{dx}(\frac{y}{x}) = e^x \) with respect to \( x \). The integral of the left side is \( \frac{y}{x} \), and the integral of \( e^x \) is \( e^x + C \), where \( C \) is an integration constant. Thus, \( \frac{y}{x} = e^x + C \).

Solve for \( y \) in Terms of \( x \)

Multiply both sides of \( \frac{y}{x} = e^x + C \) by \( x \) to solve for \( y \). This yields \( y = xe^x + Cx \).

Key concepts

First-order Linear Differential Equation

A first-order linear differential equation is a fundamental concept in calculus. It forms the building block for understanding how variables relate to their rates of change. The standard form for this equation is \( \frac{dy}{dx} + P(x)y = Q(x) \).
In this configuration, \( \frac{dy}{dx} \) represents the derivative of \( y \) with respect to \( x \). Both \( P(x) \) and \( Q(x) \) are functions of \( x \). These equations are called 'first-order' because they involve only first derivatives.

When identifying a first-order linear differential equation, it's important to look for its structure:

• The presence of \( \frac{dy}{dx} \)
• The terms \( P(x)y \) and \( Q(x) \) just involving the function itself or independent terms of \( x \).

These equations are solvable by methods such as the integrating factor, making them approachable for foundational calculus problems.

Integrating Factor

The integrating factor is a powerful technique to solve linear differential equations. It simplifies the process by transforming the equation into a form that can be easily integrated.

The integrating factor \( IF \) is found using the formula \( IF = e^{\int P(x) \, dx} \). By multiplying the entire differential equation by this factor, it becomes easier to solve. This method transforms the left side into a derivative of a product of functions, allowing straightforward integration.

Finding the integrating factor involves these key steps:

• Identify \( P(x) \) from the standard form equation: \( \frac{dy}{dx} + P(x)y = Q(x) \).
• Compute the integral of \( P(x) \), i.e., \( \int P(x) \, dx \).
• Calculate \( e^{\int P(x) \, dx} \) to get the integrating factor.

This method ensures that the original complex equation becomes something manageable, paving the way towards integration.

Integration Constant

The integration constant, commonly denoted as \( C \), arises during the process of integration. It represents an arbitrary constant that must be added when integrating functions without specific boundary conditions.

When solving first-order linear differential equations, integration is a crucial step. The integration constant acknowledges that there are infinitely many solutions to differential equations, each corresponding to different initial conditions.

Considerations for the integration constant include:

• Each indefinite integral introduces a constant \( C \).
• It's essential to apply initial conditions or additional information to determine \( C \) uniquely.
• In problems where initial values aren't given, \( C \) remains a symbolic representation of a family of solutions.

The integration constant is vital for ensuring that solutions are as general as possible until further specifics narrow them down.

Separation of Variables

Separation of Variables is another method used to solve differential equations, though not typically for linear ones. It's a technique based on isolating different variables on opposite sides of the equation, allowing each to be integrated independently.

While this method was not used directly in the provided exercise, it's effective for solving separable equations of the form \( g(y)dy = f(x)dx \).

The process of separation involves:

• Altering the equation so that all \( y \)-terms are on one side (usually with \( dy \)), and all \( x \)-terms on the other (with \( dx \)).
• Integrating both sides separately to find the general solution.

Though not applicable to the first-order linear differential equation discussed, understanding this method expands one's ability to solve a wide variety of differential equations.