Question

Solve the following differential equations by using integrating factors. $$ x y^{\prime}=x+y $$

Short answer

The solution is \( y = x \ln|x| + Cx \).

Step-by-step solution

Write The Equation In Standard Form

First, divide the entire equation by \( x \) to transform it into the standard linear form: \( y' + P(x) y = Q(x) \). Given equation: \( x y' = x + y \). Divide each term by \( x \): \( y' - \frac{1}{x} y = 1 \).

Identify P(x) and Compute The Integrating Factor

Identify \( P(x) = -\frac{1}{x} \). The integrating factor \( \mu(x) \) is given by \( e^{\int P(x) \, dx} \). Calculate \( \mu(x) = e^{\int -\frac{1}{x} \, dx} = e^{-\ln|x|} = \frac{1}{x} \).

Multiply Through By The Integrating Factor

Multiply the entire differential equation by the integrating factor \( \frac{1}{x} \): \( \frac{1}{x} y' - \frac{1}{x^2} y = \frac{1}{x} \).

Simplify Left Side Using Integrating Factor

Notice the left side simplifies to the derivative of \( \frac{y}{x} \) because \( \frac{d}{dx}(\frac{y}{x}) = \frac{1}{x} y' - \frac{y}{x^2} \). Thus, the equation becomes: \( \frac{d}{dx}(\frac{y}{x}) = \frac{1}{x} \).

Integrate Both Sides To Find y

Integrate both sides with respect to \( x \): \[ \int \frac{d}{dx}(\frac{y}{x}) \, dx = \int \frac{1}{x} \, dx \]. This gives: \( \frac{y}{x} = \ln|x| + C \).

Solve For y

Multiply through by \( x \) to solve for \( y \): \( y = x \ln|x| + Cx \), where \( C \) is the constant of integration.

Key concepts

Differential Equations

Differential equations involve derivatives and functions and are crucial in modeling situations where change occurs. They can represent anything from the movement of planets to the growth of populations. Think of them as equations that show how a rate of change in one variable affects another variable. For example, in our original exercise, the equation \( x y^{\prime} = x + y \) represents a relationship between the rate of change \( y^{\prime} \) and the variables \( x \) and \( y \). This equation is in the form of a differential equation because it includes \( y^{\prime} \), the derivative of \( y \). Understanding these equations is key to solving practical problems in physics, engineering, biology, and many other fields.

Linear Differential Equation

A linear differential equation is a type where the differential equation can be written such that each term is either a constant or the product of a constant and a function of the variable. The equation is linear in the unknown function and its derivatives. In the context of our exercise, after transforming the initial problem, we get \( y' - \frac{1}{x} y = 1 \), which fits the form of a first-order linear differential equation. Key points to remember about linear differential equations are:

• They may only involve the function and its first derivative (for first-order cases).
• They are solvable using specific methods like integrating factors.

Linear differential equations are foundational for more complex equations encountered in various scientific areas.

Standard Form Differential Equation

Standard form is an organized way to write a linear differential equation. For a first-order linear differential equation, this form is \( y' + P(x)y = Q(x) \). Writing an equation in standard form helps identify the necessary components for solving it, such as the function \( P(x) \) and the function \( Q(x) \). In the original step-by-step solution, we first rearranged the given equation \( x y^{\prime} = x + y \) to the standard form by dividing each term by \( x \) to find \( y' - \frac{1}{x} y = 1 \). This form signals us to look for the integrating factor, which aids in solving the equation. Having the equation in this format simplifies finding solutions because it leads directly to recognizable methods.

Integrating Factor Method

The integrating factor method is a powerful technique for solving linear differential equations when they are in standard form. The idea is to multiply every term in the equation by a function called an integrating factor so the equation becomes easier to integrate. The integrating factor \( \mu(x) \) is calculated using the expression \( e^{\int P(x) \, dx} \).

• First, identify \( P(x) \) from the standard form (in our equation, \( P(x) = -\frac{1}{x} \)).
• Compute the integrating factor: \( \mu(x) = e^{\int -\frac{1}{x} \, dx} = \frac{1}{x} \).

Once we multiply the equation by \( \frac{1}{x} \), it collapses into a simpler form: the left side becomes the derivative of \( \frac{y}{x} \). This allows us to solve the differential equation by straightforward integration, eventually leading to the solution \( y = x \ln|x| + Cx \). Using integrating factors turns a complex differential equation into a problem that can be handled with basic integration, making it an essential method in differential equations.