91Ó°ÊÓ

Separable equations are a type of differential equation that can be rearranged to be written as two separate parts, one involving only the dependent variable and the other involving only the independent variable. This is accomplished by separating variables. It makes integrating them more straightforward.
In our exercise, the original equation is given as \(x y^{\prime}=y\). To make it separable, we rearrange it to \(\frac{y}{x} = y^{\prime}\), which can further be expressed in differential form as \(\frac{y}{x} dx = dy\). By doing this, we isolate terms with \(y\) on one side and terms with \(x\) on the other side, which is the hallmark of separable equations.
This manipulation sets the stage for solving the differential equation through integration.

Integration is a mathematical process used to find antiderivatives. It essentially reverses differentiation and is key in solving separable differential equations.
Once a differential equation is expressed in separated form, each side is integrated with respect to its own variable. In the step-by-step solution, we integrate the left side \(\int \frac{y}{x} dx\) and the right side \(\int dy\).

• The integral \(\int \frac{y}{x} dx\) translates into \(\ln |x|\), reflecting the natural logarithm of the absolute value of \(x\).
• Meanwhile, the integral \(\int dy\) is straightforward, resulting in \(y\).

These integrals help bridge the gap between the differential form back to an algebraic expression, moving us closer to finding the equation's general solution.

The general solution of a differential equation represents a family of functions that satisfy the differential equation. It often includes an arbitrary constant, known as the constant of integration.
In our exercise, after integrating both sides, we found that \(y = \ln |x| + C\), where \(C\) represents this constant.

• The presence of \(C\) indicates that there are infinitely many solutions to the differential equation, signifying a family of curves rather than a single curve.
• This constant can be determined if additional conditions, such as initial values, are provided.

Thus, the general solution allows us to understand how the solutions vary depending on the initial conditions or additional constraints, providing a comprehensive view of the differential systems in question.