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A separable differential equation is a type of ordinary differential equation that can be rearranged so that all terms containing the dependent variable (usually denoted as \(y\)) are on one side, while all terms containing the independent variable (such as \(x\)) are on the other side. This form allows us to solve the equation through integration. The key idea is to "separate" the variables, making it easier to integrate.When faced with a differential equation like \( y' = \frac{y}{2x} \), we recognize it as separable because it can be written as \( \frac{dy}{y} = \frac{1}{2x}dx \). Separating variables is a powerful technique that transforms a differential equation into integrable form. Here are the steps:

• Identify if the equation can be rearranged into a separable form.
• Move terms to separate sides of the equation based on their variables.
• Integrate both sides separately to find the solution.

Separating the equation is a bit like unmixing a cake batter—each ingredient (or variable) has its place, making the process of solving straightforward.

The particular solution to a differential equation is a unique solution that satisfies not only the differential equation itself but also an initial condition or specific point. In simpler terms, while a general solution contains arbitrary constants, a particular solution provides a specific value for these constants based on given conditions.For instance, in our exercise, a particular solution was sought that passes through the point \((2,1)\). The general solution was found to be \( y = C'x^{1/2} \), but to find the particular solution we substituted \(x = 2\) and \(y = 1\), solving for \(C'\). Here’s more on how it works:

• Find the general solution first.
• Use the given point or condition to substitute into the general solution.
• Calculate the value of the constant to derive the particular solution.

So, the particular solution satisfies not just the underlying relationship itself, but also additional "requirements" like passing through a specific point on its curve.

The constant of integration, often denoted as \(C\), appears when solving a differential equation through integration. Since indefinite integration can have any one of infinitely many possible constants, \(C\) represents this family of possible solutions.In the context of our separable differential equation, after integrating, we found an expression involving \( \ln|y| = \frac{1}{2}\ln|x| + C \). This \(C\) stands for the general unknown constant that arises during integration.Here's a deeper look:

• Every time you integrate, a constant of integration is added because the derivative of a constant is zero—it "could" have been there all along.
• The constant remains throughout the solution until a particular condition or additional information (an initial condition, for example) allows us to solve specifically for \( C \).
• Once \(C\) is determined, through initial conditions, it becomes part of the particular solution.

The constant of integration reminds us of the flexibility of mathematical functions—their ability to shift vertically on a graph depending on initial conditions or constraints.