Question

Finding a General Solution In Exercises \(41-52,\) use integration to find a general solution of the differential equation. $$ \frac{d y}{d x}=\frac{x-2}{x} $$

Short answer

The general solution of the differential equation \(\frac{d y}{d x}=\frac{x-2}{x}\) is \(y = x - 2ln|x| + C\), where 'C' is the constant of integration.

Step-by-step solution

Separation of Variables

We start by separating the variables, moving all the terms involving y to one side, and x to the other side of the equation. This gives us: \[dy = \frac{x-2}{x} dx\] When you do this, make sure all the terms are properly rearranged so you can integrate each side correctly. This technique is based on the presumption that if two things are equal, their integrals will also be equal.

Integrate Both Sides

The following step is to integrate both sides of the equation: \[\int dy = \int \frac{x-2}{x} dx\]We can choose to break the right hand side into simpler forms for easy integration, hence:\[\int dy = \int dx - 2\int \frac{1}{x} dx\]

Solve the Integrals

Now, perform the integration on both sides:\[y = x - 2ln|x| + C\] where C denotes the constant of integration. This is the general solution of the given differential equation.

Key concepts

Differential Equation

A differential equation is a mathematical equation that relates some function with its derivatives. In simpler terms, it describes how a certain quantity changes and depends on other variables. These equations are fundamental in fields such as physics, engineering, and economics, among others.
For instance, in our exercise, the equation \( \frac{dy}{dx} = \frac{x-2}{x} \) is a first-order differential equation. This means it relates the first derivative of \( y \) with respect to \( x \), to the function itself. Differential equations can be used to model real-world phenomena, such as population growth, heat transfer, or the motion of fluids. Understanding how to solve these equations is crucial for scientists and engineers, as it allows them to predict future behavior based on current data.

Separation of Variables

Separation of variables is a technique used to solve certain types of differential equations. The goal is to rearrange the equation so that each variable is isolated on one side. This method is applicable when the equation can be written in the form \( g(y) \, dy = f(x) \, dx \), where the functions of \( y \) and \( x \) are clearly separated on different sides.
In our example, we first rewrite \( \frac{dy}{dx} = \frac{x-2}{x} \) by moving \( dy \) and \( dx \) to different sides: \( dy = \frac{x-2}{x} \, dx \). This is crucial because it allows us to integrate each side independently. The separation of variables simplifies solving differential equations by reducing them to more manageable forms that can be integrated directly.

General Solution

The general solution of a differential equation includes all possible solutions. It typically contains one or more arbitrary constants, which can be determined if additional information or initial conditions are provided.
When we integrate a differential equation, the result includes an integration constant \( C \). In our solution, after integrating both sides, we obtained: \( y = x - 2\ln|x| + C \). Here, \( C \) represents the constant that accounts for all possible outcomes of the original problem. Without specific initial conditions, this solution remains "general," meaning it encompasses all particular solutions relating to different starting points or conditions.

Integration Technique

Integration is a fundamental tool in calculus used to find functions given their derivatives. In the context of differential equations, integration helps find solutions by "undoing" the differentiation process.
The integration technique involves several methods to simplify and compute integrals. In our exercise, to integrate \( \int \frac{x-2}{x} \, dx \), we broke it down into simpler pieces: \( \int dx - 2\int \frac{1}{x} \, dx \). This decomposition made it straightforward to perform the integration, resulting in \( y = x - 2\ln|x| + C \). Understanding how to manipulate the integrands and apply integration rules efficiently is key to finding solutions to differential equations effectively.