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Separable differential equations are a type of differential equation that can be manipulated such that all terms involving the dependent variable (usually denoted as \( y \)) are on one side of the equation, and all terms involving the independent variable (usually \( x \)) are on the other side. This "separation" allows for the equation to be solved through integration.
The essential form of a separable differential equation is:

• \( y' = g(x)h(y) \)

To solve a separable differential equation, you rewrite it as:

• \( \frac{dy}{h(y)} = g(x) \, dx \)

By separating the variables, we can integrate both sides individually, which is a significant step toward finding the solution. This method is powerful for solving many first-order differential equations because it breaks down complex problems into simpler integrations.

Integration by separation is the process used after rearranging a separable differential equation into a form where each side can be integrated independently of the other. This process involves two primary steps:
1. **Separate** the variables: Rewrite the equation so that all instances of the dependent variable \( y \) are on one side and all instances of the independent variable \( x \) are on the other:

• \( \frac{dy}{y^2} = 3(x + \cos x) \, dx \)

2. **Integrate** both sides: Once separated, each side of the equation is integrated with respect to its own variable:

• \( \int \frac{1}{y^2} \, dy = \int 3(x + \cos x) \, dx \)

The left side simplifies to \(-\frac{1}{y}\), while the right side becomes \(\frac{3x^2}{2} + 3\sin x + C\). This step reduces the equation to a solvable form by applying basic integration rules.

A first-order differential equation is a relation involving the derivative \( y' \), the function \( y \) itself, and the independent variable \( x \). These equations are termed "first-order" because they only involve the first derivative and do not include higher-order derivatives like \( y'' \) or \( y''' \).
In our problem example, we are dealing with a first-order equation \( y' = 3y^2(x + \cos x) \), where \( y' \) signifies the rate of change of the function \( y \) with respect to \( x \). The primary goal is to determine how this rate of change results in a specific behavior of \( y \) over a domain of \( x \). Recognizing the equation as first-order helps decide the suitable methods for solving it, like separation of variables in this case.

After finding the general solution of a differential equation through separation and integration, applying initial conditions allows us to find a particular solution that fulfills a given condition at a specific point. An initial condition specifies the value of the solution at a particular point, usually written as \( y(x_0) = y_0 \).
For the equation \( y' = 3y^2(x + \cos x) \), we apply the initial condition \( y(0) = -2 \) after we've obtained the general solution. This process involves substituting \( x = 0 \) and \( y = -2 \) into the solution:

• \( -2 = -\frac{1}{\frac{3 \times 0^2}{2} + 3\sin(0) + C} \)

Solving for the constant \( C \), in this case, gives \( C = \frac{1}{2} \). Applying initial conditions ensures our solution precisely fits the model described by the problem and accounts for any initial behavior exhibited by the system.