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First-order differential equations are equations involving the first derivative of a function. These types of equations are fundamental in mathematical analysis because they describe how a function changes intuitively. Typically, a first-order differential equation appears in the form \(\frac{dy}{dx} = f(x, y)\), where \(y\) and \(x\) are variables, and \(f(x, y)\) is a given function describing the relationship between them.
For many real-world situations, such as in physics or population dynamics, these equations model changing rates. They provide a way to predict future behavior of systems by looking at the current rate of change. Understanding these equations is crucial for interpreting various natural and social phenomena.

Integration is a mathematical process used to find a function from its derivative, which in the context of differential equations, is crucial for solving them. When we integrate a differential equation like \(\frac{dy}{y^2} = \int \frac{dx}{x^2}\), we essentially reverse the differentiation to find the original function, \(y(x)\).
This process provides a way to determine cumulative quantities, such as area under a curve or the total change across an interval for a given rate of change. Integration requires applying rules and techniques, like substitution or partial fractions, to evaluate integrals correctly. Understanding integration opens doors to solving more complex equations and predicting system behaviors over time.

Variable separation is a technique used to solve differential equations by rearranging them so that each variable is on its own side of the equation. This is particularly applicable to separable differential equations, where you can rearrange terms to isolate \(y\) and its derivatives from \(x\) and its derivatives.
For instance, in the equation \(\frac{dy}{y^2} = \frac{dx}{x^2}\), we separate the variables to integrate them easily. This requires rearranging terms, allowing us to solve the equation by integrating each part independently. Mastering this method simplifies solving differential equations and makes complex problems more manageable.

The general solution of a differential equation provides a family of functions that satisfy the equation. It includes an arbitrary constant, often denoted as \(C\), which can be adjusted based on initial or boundary conditions specific to a problem.
In the example, after integration, the general solution expressed is \(y(x) = \frac{x}{1 - Cx}\). This equation describes all possible curves that solve the differential equation. The solution becomes specific when a value for \(C\) is known from given conditions. Understanding the general solution is essential for applications in engineering, physics, or any field requiring differential equation modeling.