91Ó°ÊÓ

In the world of differential equations, a 'separable differential equation' is often a straightforward form that’s relatively simple to solve. Such equations can be rewritten so that each of the variables appears on opposite sides of the equation. This feature is particularly helpful, as it allows for the integration process to occur separately for each variable.
The key idea in solving these equations is to separate the variables, such as having all terms involving \(y\) and its derivatives on one side, and all terms involving \(t\) on the other.

• For example, consider the differential equation \(\frac{d y}{d t}=\frac{4 y}{5}-\frac{y^{2}}{150}\).
• The goal is to rearrange it to make \(\frac{dy}{\left(\frac{4 y}{5}-\frac{y^{2}}{150}\right)} = dt\), where the separation of variables has occurred.

Once in this form, both sides of the equation can be integrated, which allows us to solve for the function \(y(t)\). This method of separation is a powerful tool that simplifies the process of solving complex differential equations.

Integration forms the heart of solving a differential equation once the variables have been separated. When we integrate these separable differential equations, we are essentially peeling back layers to uncover the function \(y(t)\).
With the separated equation from the example, \(\int \frac{1}{\left(\frac{4 y}{5}-\frac{y^{2}}{150}\right)} dy = \int dt\), we integrate both sides to transform it from a differential equation to a more familiar algebraic one.

• On the left side, the integration is with respect to \(y\), which usually involves finding an antiderivative or using methods such as partial fraction decomposition if necessary.
• The right side integration is often simpler, directly resulting in \(t + C\), where \(C\) is the integration constant.

Through this process, the solution emerges as a relationship between \(y\) and \(t\), unveiling the function that describes the behavior of the original differential equation over time.

The constant of integration, represented by \(C\), plays a crucial role when solving differential equations. It appears naturally as a result of the indefinite integration process.
In the context of the logistic differential equation problem, after integrating, we arrive at an expression like \(-150 \ln (| 30 - y |) + 750 \ln (| y |) = 4t + C\).

• Initially, \(C\) is a placeholder for any real number since indefinite integrals can have an infinite number of solutions.
• Its actual value is determined by the initial condition given with the problem, which ensures that the solution fits the situation described.

In our example, the solution at \(t = 0\) and \(y = 8\) is used to calculate \(C\) as \(-963.744\). This adjustment ensures the solution precisely fits the point through which the differential equation is required to pass. Understanding \(C\) helps not just in finding the specific solution we are seeking, but also in appreciating the infinite possibilities an integration process inherently provides.