91Ó°ÊÓ

Ordinary differential equations (ODEs) are equations that involve a function of one independent variable and its derivatives. These types of equations are immensely significant in mathematics and science, as they often model the behavior of physical systems. ODEs are termed 'ordinary' to distinguish them from partial differential equations, which involve multiple independent variables.
In the context of ODEs, a common goal is to find a function that satisfies the given differential equation. The equation in our example is a first-order ODE. The "first-order" indicates that the highest derivative present is the first derivative, written as \( \frac{dy}{dx} \). The aim is to find the general form of the function \( y(x) \).
Overall, solving ODEs can range from fairly straightforward to quite complex, depending on the equation's nature. For first-order ODEs, techniques like separation of variables are often applicable, which we will explore in the next section.

Separation of variables is a powerful method for solving ordinary differential equations when the equation can be expressed such that each variable appears on a separate side of the equation. It's especially useful for first-order ODEs like the one we have:
\[ \frac{dy}{dx} = \frac{1}{1-x} \]
The primary goal is to "separate" the equation so that one side only contains terms related to \(y\) and the other side only terms related to \(x\). In this case, you multiply both sides by \(dx\) to achieve this separation:
\[ dy = \frac{1}{1-x} \ dx \]
By doing so, you've created an equation that can be integrated directly. Separation of variables requires that both sides of the equation be integrable, which means you can compute the integral of each side independently. This will then lead you to the solution of the differential equation.

Integration is the process of finding a function from its derivative, which forms a cornerstone in solving equations through separation of variables. Once the variables are neatly separated in the equation \( dy = \frac{1}{1-x} \, dx \), you integrate both sides:
\[ \int dy = \int \frac{1}{1-x} \, dx \]
The left side, \( \int dy \), is straightforward and equates to \( y + C_1 \), where \( C_1 \) is an arbitrary constant. The right side involves integrating \( \frac{1}{1-x} \), a well-known result that gives \( -\ln|1-x| + C_2 \).
Thus, equating both sides gives you:
\[ y + C_1 = -\ln|1-x| + C_2 \]
By consolidating the constants \(C_1\) and \(C_2\) into a single constant \(C\), and considering that \(x > 1\) in the problem statement, you can remove the absolute value bars and write the simplified solution as:
\[ y = -\ln(x-1) + C \]
Integration, in this context, transforms the differential equation into an algebraic one by solving for \(y\), illustrating its critical role in finding solutions to differential equations.