91Ó°ÊÓ

Separable differential equations are a special class of differential equations that can be broken down into the separate functions of two independent variables. These equations are particularly user-friendly because they allow both sides to be integrated independently.
For a differential equation to be separable, it should be possible to rearrange the equation into the form \( \frac{dy}{dx} = g(x)h(y) \). This means that you can express it as a product of a function of \( x \) and a function of \( y \). Essentially, you want to move all terms involving \( y \) to one side and all terms involving \( x \) to the other side.
In the problem we have, \( y' = \frac{y}{x} \) is indeed separable. By recognizing that we can rewrite this equation as \( \frac{dy}{dx} = \frac{y}{x} \), we see that it fits the pattern, which then allows us to proceed to the next steps of solving it.

In the context of separable differential equations, integration techniques play a crucial role in finding a solution. After separating the variables, the next step involves integrating each side of the equation.
For our problem, after separation, we have \( \frac{dy}{y} = \frac{dx}{x} \). This is where the power of integration comes into play. Integrating the left-hand side, \( \int \frac{dy}{y} \), results in the natural logarithm, \( \ln |y| \). Similarly, integrating \( \int \frac{dx}{x} \) on the right-hand side gives us \( \ln |x| + C \), where \( C \) is the constant of integration.
Only through integration can we transition from a differential equation to an equation involving no derivatives, thereby getting one step closer to an explicit solution for \( y \). This illustrates why mastering integration techniques is fundamental for solving differential equations.

A general solution of a differential equation is a solution that includes an arbitrary constant. This constant represents an infinite number of possible solutions.
In solving separable differential equations like \( y' = \frac{y}{x} \), after integration, we arrive at \( \ln |y| = \ln |x| + C \). The next step is to solve for \( y \), which involves removing the logarithmic function by exponentiation.
Exponentiating both sides, we have \( e^{\ln |y|} = e^{\ln |x| + C} \). This simplifies to \( |y| = e^{C} |x| \), hence \( y = \pm e^{C} x \). To simplify, we often represent \( e^{C} \) by another constant, say \( C' \), leading to a general solution of the form \( y = C'x \).
This form of solution provides immense flexibility because varying \( C' \) gives us a family of functions, all of which satisfy the original differential equation. Thus, the concept of a general solution is at the heart of understanding differential equations and represents a crucial step in their analysis.