91Ó°ÊÓ

The technique of separating variables is a fundamental method used to solve differential equations. Especially useful in solving autonomous differential equations, this method works by rearranging the equation so that each variable appears on a separate side of the equation. In our problem with the differential equation \(\frac{dN}{dt} = 5 - N\), separation of variables involves rewriting it as \(\frac{dN}{5-N} = dt\). This structure allows us to integrate both sides independently, which is crucial for solving the equation step by step.

By separating the variables, we isolate \(N\)-related terms from \(t\)-related ones. This process prepares the equation for integration. Once separated, the differential equation becomes easier to handle, making it possible to find a solution by subsequent integration.

The initial condition in a differential equation provides essential information needed to find a particular solution. It defines a specific point in time (or space) where the value of the solution is known. Initial conditions are critically important because many differential equations have an infinite number of solutions; the initial condition helps pin down which one is relevant to the given situation.

In our example, the initial condition is given as \(N(2) = 2\). This means that when \(t = 2\), the value of \(N\) is precisely 2. By using this information, we can determine the integration constant after performing the integration step. By substituting \(t=2\) and \(N=2\) into the integrated equation, we solve for the constant that makes the equation conform to this specific solution behavior.

A particular solution of a differential equation is a solution that satisfies both the differential equation and the initial conditions. It is not just any solution; it is the specific one that fits within the unique constraints of the problem.

For our autonomous differential equation, after performing integration and incorporating the initial condition, we derive the particular solution. In this case, the solution \(N = 5 - 3e^{2-t}\) fits both the differential relationship and the initial condition. This solution perfectly describes how the quantity \(N\) evolves over time \(t\) in alignment with the initial condition when \(t=2, N=2\).

• It provides not just any trajectory but the precise path \(N\) follows under the given circumstances.
• This particular solution is vital because it ensures that the modeled behavior accurately reflects the real-world system described by the initial condition.

The integration constant, represented often by \(C\), emerges naturally from the process of integrating a differential equation. Since indefinite integration results in a family of curves or functions differing by a constant, we introduce \(C\) to account for this variability.

In solving \(\int \frac{1}{5-N} \, dN = \int 1 \, dt\), following integration, an expression involving \(C\) appears: \(-\ln|5-N| = t + C\). The role of \(C\) is crucial because it allows us to modify the generic solution such that it passes through the specific initial condition provided.

Using the initial condition \(N(2) = 2\), we calculated \(C\) as \(-\ln 3 - 2\). This calculation is a key step in personalizing the solution, ensuring it is not just a generic solution but one tailored exactly to fit the given conditions defined by the problem.