Question

Solve each differential equation. $$ \frac{d y}{d x}+\frac{y}{x}=\frac{1}{x} $$

Short answer

The solution is \( y = 1 + \frac{C}{x} \).

Step-by-step solution

Recognize the Type of Differential Equation

This is a linear first-order differential equation of the form \( \frac{d y}{d x} + P(x) y = Q(x) \). Specifically, here, \( P(x) = \frac{1}{x} \) and \( Q(x) = \frac{1}{x} \).

Find the Integrating Factor

The integrating factor \( \mu(x) \) is given by \( e^{\int P(x) \, dx} \). Calculate this integral: \( \int \frac{1}{x} \, dx = \ln|x| \). Thus, the integrating factor \( \mu(x) = e^{\ln|x|} = |x| \). Since \( x > 0 \) (implied context), we use \( x \).

Multiply the Differential Equation by the Integrating Factor

Multiply both sides of the differential equation by \( x \) to get \( x \frac{d y}{d x} + y = 1 \). This transforms our original equation into an exact equation in terms of derivatives.

Solve the Original Differential Equation

Notice that \( \frac{d}{dx}(xy) = x \frac{dy}{dx} + y \). Hence, the equation \( x \frac{d y}{d x} + y = 1 \) can be rewritten as \( \frac{d}{dx}(xy) = 1 \).

Integrate Both Sides

Integrate \( \frac{d}{dx}(xy) = 1 \) with respect to \( x \). The left hand side integrates to \( xy \) and the integral of 1 is \( x + C \), where \( C \) is the constant of integration. Thus, \( xy = x + C \).

Solve for y

Divide both sides by \( x \) to solve for \( y \): \( y = 1 + \frac{C}{x} \). This is the general solution of the differential equation.

Key concepts

First-Order Differential Equations

First-order differential equations are equations that involve derivatives of a function with respect to one variable, where the highest derivative is of the first order. This means that the equation contains terms like \( \frac{dy}{dx} \), but not terms like \( \frac{d^2y}{dx^2} \). These equations are generally written in the form \( \frac{dy}{dx} + P(x)y = Q(x) \). They describe a wide variety of natural phenomena and are fundamental in understanding how different quantities change over time or space.

To solve these equations, one often uses specific techniques based on the structure of the equation. Recognizing the equation's type is crucial for selecting the appropriate method for finding a solution.

Linear Differential Equations

A linear differential equation is a type of first-order differential equation where the unknown function, \( y \), and its derivatives appear linearly. This means that every term involving \( y \) or its derivatives is of the first degree or is multiplied by a function of \( x \) rather than another term in \( y \).

The standard form of a linear differential equation is \( \frac{dy}{dx} + P(x)y = Q(x) \). The key to solving these equations lies in their linearity, which allows the use of an integrating factor to simplify the equation and find a solution. This makes linear differential equations particularly manageable compared to other types of differential equations.

Integrating Factor

The integrating factor is a powerful tool used to solve first-order linear differential equations. It is a function, usually denoted as \( \mu(x) \), that transforms a non-exact differential equation into an exact equation. This makes the equation solvable with straightforward integration.

To find the integrating factor, calculate \( e^{\int P(x) \, dx} \). For the equation \( \frac{dy}{dx} + \frac{y}{x} = \frac{1}{x} \), the integrating factor calculated is \( x \), because \( \int \frac{1}{x} \, dx = \ln|x| \) and consequently, \( \mu(x) = e^{\ln|x|} = x \).

• Multiply through the differential equation by this integrating factor.
• The equation becomes \( \frac{d}{dx}(xy) = 1 \), a perfect derivative.

This transformed equation can be easily integrated, leading us closer to the general solution.

General Solution

The general solution of a differential equation gives all possible solutions that satisfy the equation. For a first-order linear differential equation, the general solution is typically expressed in terms of an arbitrary constant, \( C \).

In the context of our example, once the integrating factor simplifies the problem to \( \frac{d}{dx}(xy) = 1 \), integrating both sides with respect to \( x \) gives \( xy = x + C \). Solving for \( y \) yields the general solution: \( y = 1 + \frac{C}{x} \).

• This solution represents a family of curves that describe all possible behaviors of the system described by the differential equation.
• By choosing different values for \( C \), you find particular solutions that correspond to specific initial conditions or constraints.

Understanding the general solution allows for a comprehensive view of the problem's solution space.