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Magazine Chapter 4: Problem 228
Solve the following differential equations by using integrating factors. $$ x y^{\prime}=x+y $$
Short Answer
Expert verified
The solution is \( y = x(\ln|x| + C) \).
Step by step solution
01
Rearrange the Equation
The first step is to rearrange the given differential equation into the standard form of a linear first-order differential equation, which is \( y^{\prime} + P(x)y = Q(x) \). Starting from the given equation \( xy^{\prime} = x + y \), divide each term by \( x \) to obtain \( y^{\prime} - \frac{1}{x}y = 1 \). Now we have it in the standard form with \( P(x) = -\frac{1}{x} \) and \( Q(x) = 1 \).
02
Find the Integrating Factor
Next, calculate the integrating factor, \( \mu(x) \), which is given by \( e^{\int P(x) \, dx} \). Since \( P(x) = -\frac{1}{x} \), we integrate it to find \( \int -\frac{1}{x} \, dx = -\ln|x| \). The integrating factor is then \( \mu(x) = e^{-\ln|x|}= \frac{1}{x} \).
03
Multiply Through by the Integrating Factor
Multiply the entire differential equation by the integrating factor \( \frac{1}{x} \). This gives us \( \frac{1}{x}y^{\prime} - \frac{1}{x^2}y = \frac{1}{x} \).
04
Recognize as a Derivative
Notice that the left-hand side of the equation can be written as the derivative of a product, \( \frac{d}{dx}(\frac{1}{x}y) \). So the equation becomes \( \frac{d}{dx}(\frac{1}{x}y) = \frac{1}{x} \).
05
Integrate Both Sides
Integrate both sides of the equation with respect to \( x \). The left-hand side integrates to \( \frac{1}{x}y = \int \frac{1}{x} \, dx = \ln|x| + C \), where \( C \) is the constant of integration.
06
Solve for y
Finally, solve for \( y \) by multiplying both sides by \( x \), resulting in \( y = x(\ln|x| + C) \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integrating Factor
The integrating factor is a crucial tool used in solving linear first-order differential equations. It transforms a differential equation into a form that can be easily integrated.
To find the integrating factor, we start with a differential equation in the form of \( y' + P(x)y = Q(x) \). The integrating factor \( \mu(x) \) is given by the expression \( e^{\int P(x) \ dx} \). This factor simplifies the process by making the left-hand side of the differential equation a perfect derivative.
To find the integrating factor, we start with a differential equation in the form of \( y' + P(x)y = Q(x) \). The integrating factor \( \mu(x) \) is given by the expression \( e^{\int P(x) \ dx} \). This factor simplifies the process by making the left-hand side of the differential equation a perfect derivative.
- Calculate \( \mu(x) \) by integrating \( P(x) \) and exponentiating the result.
- Multiply the entire differential equation by \( \mu(x) \).
First-Order Differential Equation
A first-order differential equation is one that relates a function to its first derivative. These types of equations are fundamental in understanding various processes that change over time, such as population growth or cooling of a liquid.
The general form of a first-order differential equation is \( y' = f(x, y) \), where \( y' \) denotes the derivative of \( y \) with respect to \( x \). Solutions to these equations provide a function \( y(x) \) that satisfies the given relationship with its derivative.
The general form of a first-order differential equation is \( y' = f(x, y) \), where \( y' \) denotes the derivative of \( y \) with respect to \( x \). Solutions to these equations provide a function \( y(x) \) that satisfies the given relationship with its derivative.
- They are called 'first-order' because the highest derivative involved is the first.
- Such equations can often be solved using separation of variables, integrating factors, or substitutions.
Linear Differential Equation
Linear differential equations are a subtype of differential equations where the unknown function and its derivatives appear linearly. These equations can be represented in a standard way that helps in their resolution.
The form \( y' + P(x)y = Q(x) \) is typical for a first-order linear differential equation. Key aspects include:
The form \( y' + P(x)y = Q(x) \) is typical for a first-order linear differential equation. Key aspects include:
- No powers or products of the function \( y \) or its derivative \( y' \) appear.
- The equation is linear in terms of both \( y \) and \( y' \).
