91Ó°ÊÓ

Antiderivatives are functions that reverse the process of differentiation. If you have a function, and you take its derivative to find its rate of change, the antiderivative is what you would integrate to return to the original function. This makes antiderivatives crucial when solving differential equations. In the context of our given problem, we see that the differential equation is:
\[\frac{dy}{dx} = \frac{-1}{x}\]To solve this using antiderivatives, we integrate both sides. By integrating, we essentially find a function whose derivative gives the function we started with:

• The left side becomes \( y \).
• The right side, upon integration, results in \( -\ln|x| \).

So, the solution we get is:\[y = -\ln|x| + C\]where \( C \) is a constant of integration, representing the family of solutions that fulfill different initial conditions.

Separation of variables is a method used to solve certain types of first-order differential equations. This approach splits the variables in the differential equation, allowing you to integrate each one separately. However, in our problem stance, the equation
\[\frac{dy}{dx} = \frac{-1}{x}\]does not require this technique because the right-hand side is only a function of \( x \). This makes the separation of variables unnecessary in this instance, but it’s crucial to recognize when it can be useful.
In general, separation of variables works when you can rearrange the equation into the form:

• \( f(y) dy = g(x) dx \)

Once separated, each side can be integrated independently, which is a powerful method for finding solutions when both \( x \) and \( y \) are intertwined.

First-order differential equations involve derivatives of a function in terms of a single variable and can often be identified by their form \( \frac{dy}{dx} = f(x, y) \). They play a significant role in various fields like physics, engineering, and economics.
Our problem is a prime example:
\[\frac{dy}{dx} = \frac{-1}{x}\]This is a first-order differential equation because it involves the first derivative \( \frac{dy}{dx} \). Depending on its form, different methods are used to solve such equations, such as:

• Using antiderivatives, which we applied in our problem to find \( y = -\ln|x| + C \).
• Using separation of variables for equations where both variables can be separated appropriately.

First-order differential equations are foundational because they offer the simplest description of how a quantity changes in relation to another.