Question

Solving a First-Order Linear Differential Equation In Exercises \(5-14,\) solve the first-order linear differential equation. $$ (x-1) y^{\prime}+y=x^{2}-1 $$

Short answer

The solution to the differential equation \((x-1) y' + y = x^2 -1 \) is \(y = 1/3*x^2 -1 + C/x\) where C is a constant of integration.

Step-by-step solution

Standardize the Equation

The first step involves rewriting the given equation \((x-1) y'+y=x^2-1\) making y' the subject. Thus, we obtain \(y'+(1/(x-1))y=(x+1)/(x-1)\).

Calculate the Integrating Factor

Next, we calculate the integrating factor which is \(e^{\int P(x) dx}\). Since \(P(x)=1/(x-1)\), the integrating factor becomes \(e^{ \int (1/(x-1)) dx}\) which simplifies to \(x-1\).

Substitute the Integrating Factor

The third step is to multiply the whole equation by the integrating factor. Multiplying \(x-1\) through gives the equation \(x(y'+(1/(x-1))y) = x^2-1\). Simplifying this equation yields \((x*y)' = x^2 - 1\).

Integrate the Equation

Integrating both sides of the equation gives \(\int (x*y)' dx = \int (x^2 - 1) dx\). The integral on the left simplifies to \(x*y\) and the right side becomes \(1/3*x^3 - x + C\) where C is the constant of integration. Dividing both sides by x gives \(y = 1/3*x^2 -1 + C/x\).

Key concepts

Integrating Factor

The integrating factor is a mathematical tool that makes solving first-order linear differential equations much easier. To find it, you need to focus on a specific part of the differential equation. For example, if the equation is in the form of \( y' + P(x)y = Q(x) \), your goal is to calculate the integrating factor, which is denoted as \( e^{\int P(x) dx} \). This step effectively transforms the differential equation into an easier form to integrate, allowing us to solve it step by step.

Once you have the integrating factor, you will multiply the entire differential equation by it. This action modifies the equation into a form where the left-hand side is the derivative of a product, specifically \((e^{\int P(x) dx} \, y)'\). The result simplifies the integration process in subsequent steps.

• Find \( P(x) \) from the equation.
• Calculate \( e^{\int P(x) dx} \), the integrating factor.
• Multiply the full equation by the integrating factor.

Mastering the concept of the integrating factor is crucial for anyone studying differential equations.

Standard Form

Understanding the standard form of a first-order linear differential equation is key to applying various methods for solutions. In its simplest form, this type of equation appears as \( y' + P(x)y = Q(x) \). This format allows you to easily identify the functions \( P(x) \) and \( Q(x) \), which are essential for finding the integrating factor and subsequently solving the equation.

To convert any given differential equation into its standard form, you first need to rearrange terms so that the derivative \( y' \) is isolated on one side. For example, the equation \((x-1) y' + y = x^2 - 1\) can be transformed by dividing every term by \(x-1\), yielding \( y' + (1/(x-1))y = (x+1)/(x-1) \). Now, the equation is in its desired standard form, ready for the next steps in solving it.

• Rearrange the equation to isolate \( y' \).
• Divide terms if necessary to achieve the form \( y' + P(x)y = Q(x) \).

Converting to standard form is a necessary preparatory step before tackling the main solution methods.

Differential Equations

Differential equations involve derivatives and are used to describe various phenomena in engineering, physics, and other fields. A first-order linear differential equation is particularly common and involves the first derivative of a function. The general form is \( y' + P(x)y = Q(x) \). Such equations are particularly valuable as they often model real-world situations where change is a constant factor.

Solving these equations involves several systematic steps, including finding the integrating factor, transforming the equation, and using integration. These steps are consistent regardless of the specific functions involved. By mastering these techniques, students can solve a wide variety of practical problems.

• Identify the kind of differential equation you are dealing with.
• Use appropriate solution methods for first-order linear types.
• Recognize real-world applications for differential equations.

Differential equations are a gateway to understanding the dynamics underlying many scientific and engineering problems.

Constant of Integration

The constant of integration, often represented by \( C \), plays a crucial role in differential equations, particularly when calculating indefinite integrals. When you integrate a function, there are infinite possibilities for the resulting antiderivative due to the presence of this constant.

During the solving process of a differential equation, the constant of integration is introduced when you integrate both sides of an equation. It signifies that there are many potential solutions to the differential equation based on the initial conditions or boundary conditions provided in a problem. For example, when computing the integral of \( (x^2 - 1) \), you find \( 1/3 \, x^3 - x + C \) as the result, where \( C \) accounts for any constant shift in the solution.

• The constant arises from indefinite integrations.
• It represents multiple potential specific solutions.
• In problems, an initial condition might be used to determine \( C \).

Understanding the constant of integration allows for precise solutions tailored to specific conditions within a given problem.