91Ó°ÊÓ

A separable differential equation is a type of first-order differential equation that can be written in the form of `\( \frac{dy}{dx} = g(x)h(y) \)`. The beauty of this form lies in its ability to be split into two parts, one involving only \( x \) and the other only \( y \), which we can then integrate independently. This type of equation is common in real-world applications due to its simplicity and the straightforward strategy for solving it.

In the case of our differential equation, `\( \frac{d N}{d t} = \frac{1}{t} \)`, the equation is already separable. We isolate the variables so that we have all the \( N \) terms on one side and all the \( t \) terms on the other. This step essentially transforms the equation into an integrable form:

- Move \( dt \) to the other side: `\( dN = \frac{1}{t}dt \)`.
- Both sides are now ready to be integrated independently.

This separation of variables is crucial as it allows us to rewrite the equation in a way that is conducive to integration, leading us to the solution of the differential equation.

When you solve a separable differential equation by integrating, it’s important to remember the integration constant. This constant arises because indefinite integration can have multiple solutions, each differing by a constant term. Whenever we integrate a function, we can add any constant value, which is why we include +\( C \) in our integration results.

In the given problem, after integrating \( \int \frac{1}{t} \, dt \), we get \( \ln |t| + C \). Here:

- \( C \) is an unknown constant of integration.
- We initially don't know its exact value until we apply the initial condition, which helps us solve for \( C \).

In practice, the integration constant accounts for the specific solution to our initial-value problem. Without it, our solution would represent a family of infinitely many potential curves rather than the specific curve that satisfies the given initial condition.

An initial condition, in this context, is a piece of information that allows us to find the specific solution to a differential equation that would pass through a given point on its graph. It can be thought of as an anchor point.

In the problem, the initial condition is given as \( N(1) = 10 \). This condition is used to determine the value of the integration constant \( C \).

Upon reaching the solution \( N(t) = \ln |t| + C \), applying the initial condition involves:

- Substituting \( t = 1 \) into the equation:

\( N(1) = \ln |1| + C = 10 \)
- Since \( \ln 1 = 0 \), the equation simplifies to \( C = 10 \).

Thus, the initial condition allows us to pin down our particular solution to \( N(t) = \ln t + 10 \). This means that as \( t \) increases starting from \( t = 1 \), the function behavior is exactly determined by this specific formula, perfectly matching the given initial data.