91Ó°ÊÓ

Separation of variables is a fundamental method for solving differential equations. It involves rearranging the equation to isolate terms involving one variable on each side. In our example, we have the differential equation \( \frac{dy}{dx} = xy^2 \). The goal is to separate the variables \( y \) and \( x \), each with their respective differentials. To achieve this, we rearrange the equation by multiplying both sides by \( dx \) and \( y^{-2} \). This operation moves all \( y \)-related terms to one side and all \( x \)-related terms to the other, resulting in the equation \( y^{-2} \, dy = x \, dx \). Now, we are prepared to integrate each side independently, which simplifies the process and leads us to the next step.

Integration is the mathematical process of finding the antiderivative of a function. In the context of solving differential equations, integration is used to find a general solution. Once we have separated the variables, as in our example equation \( y^{-2} \, dy = x \, dx \), the next step is to integrate each side.

• For the left side, \( \int y^{-2} \, dy \), the antiderivative is \(-y^{-1} \).
• For the right side, \( \int x \, dx \), the antiderivative is \( \frac{x^2}{2} \).

By integrating, we convert the differential equation into an algebraic expression. The result of integration is: \(-y^{-1} = \frac{x^2}{2} + C \). Here, \( C \) is an arbitrary constant, which we will discuss next.

Whenever we integrate, an arbitrary constant, commonly referred to as the constant of integration, is added. This constant \( C \) reflects the infinitely many solutions that can satisfy a differential equation. After integrating each side of our equation and obtaining \(-y^{-1} = \frac{x^2}{2} + C \), the constant \( C \) becomes critical. It represents the final adjustment needed to ensure that the solution is complete.
Without \( C \), the solution would be specific to just one instance of the general solution.

• The solution to a differential equation can represent a family of curves, each shifted vertically by different values of \( C \).
• This flexibility allows us to apply any initial conditions or constraints that constrain \( C \) to a particular value.

Solving for \( y \), we manipulate the algebraic expression to reveal \( y = \frac{1}{-\frac{x^2}{2} - C} \), thereby illustrating how \( C \) affects the shape and position of the solution's curve.