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The general solution of a differential equation is an expression that includes all possible solutions of the equation. In these equations, especially in a first-order ordinary differential equation like the one we are considering, the general solution typically includes a constant that can be any real number. This constant is included because when we solve a differential equation by integration, we add a constant of integration. In this context, for the equation \( \frac{dy}{dx} = y \), the general solution is expressed as:

• \( y = Ae^x \)

Here, \( A \) represents the constant, allowing for a family of solutions that satisfy the differential equation based on different initial conditions. Understanding the concept of the general solution is crucial because it informs us of all possible outcomes that satisfy the given differential equation.

A separable differential equation is a specific type of differential equation where the variables can be separated on opposite sides of the equation. This means you can rearrange terms so that all terms containing one variable and its differential are on one side of the equation and all terms containing the other variable and its differential are on the other side.
For the given problem, the equation \( \frac{dy}{dx} = y \) can be rewritten into a separable form by moving all terms involving \( y \) to one side and all terms involving \( x \) to the other side. This results in the expression:

• \( \frac{1}{y} \, dy = dx \)

Solving separable differential equations involves integrating both sides independently. This step is possible because once we separate the variables, the equation is effectively simplified into a product of separate integrals, which leads directly into the next crucial phase: integration.

Integration is a fundamental concept in calculus and an essential step in solving differential equations, particularly when dealing with separable differential equations. It involves finding the antiderivative of a function, which essentially reverses the process of differentiation.
For a separable differential equation like \( \frac{1}{y} \, dy = dx \), the integration process involves integrating each side:

• \( \int \frac{1}{y} \, dy = \int dx \)

The result of these integrations will lead us to:
\( \ln |y| = x + C \), where \( C \) is the constant of integration. To find \( y \), we exponentiate both sides:
\( |y| = e^{x+C} \), which simplifies further to:

• \( y = Ae^x \)

Here \( A = \pm e^C \), introducing a constant that reflects the indefinite nature of integration and covering all potential solutions to our original differential equation. Understanding how to integrate each side of an equation independently is crucial for solving separable differential equations.