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A differential equation is a type of mathematical equation that involves an unknown function along with its derivatives. These equations play a critical role in various fields such as physics, engineering, and economics, as they describe various phenomena like motion, heat, or waves. In simple words, differential equations can be thought of as equations that describe change. For example, the differential equation in our exercise, \[ y' = \frac{3y}{2+x} \], describes how the function \( y \) changes with respect to \( x \).

• "Differential" refers to the derivatives, which indicate rates of change.
• "Equation" implies a mathematical expression that combines these derivatives.

Integration is the mathematical process of finding the integral of a function. It serves as the reverse operation of differentiation and is a vital tool in solving differential equations. When dealing with separable differential equations like the one we have, integration allows us to determine the function from its rate of change.

• Integration helps in accumulating quantities, such as areas under curves.
• It can also be used for determining antiderivatives in calculus.

In the given step-by-step solution, integration is utilized when both sides of the equation are separately integrated:\[ \int (2+x) \, dx = \int 3y \, dy \] This step is crucial in solving the equation and finding the unknown function \( y \).

The general solution of a differential equation encompasses all possible solutions, including arbitrary constants that might arise during integration. In the exercise, the general solution is determined after integrating both sides of the equation and solving for \( y \).
The concept of a general solution is important because different initial conditions applied to the same differential equation might yield different specific solutions. Thus, having the general solution allows us to capture all such possibilities. The exercise concludes with the general solution:\[ y = \sqrt{\frac{2}{3}(x^2 + 2x) - \frac{2}{3}C} \] Here, \( C \) is the arbitrary constant, which will account for these variations.

Separable equations are a special class of differential equations where the variables can be separated to opposite sides of the equation. This characteristic simplifies the process of finding a solution. Example of such an equation is:\[ y' = \frac{3y}{2+x} \]

• We separate the variables by moving all terms involving \( y \) to one side and terms involving \( x \) to the other:
• \( (2+x) \, dx = 3y \, dy \)

By simplifying and integrating each side, we can find the general solution of the equation. This method is powerful for solving first-order differential equations and is widely used in both academic and professional settings.