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Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting radiation. In this problem, the decay process is modeled by a differential equation, where the decay rate is proportional to the mass of the substance present. The decay rate given as \( \frac{1}{10} \) of the mass means that each hour, \( 10\% \) of the remaining radioactive substance will decay.

For example, if you start with 20 grams, after 1 hour, 2 grams (i.e., \( \frac{20}{10} \)) will decay, leaving 18 grams. This decay continues until eventually, the mass decreases significantly. Understanding this concept is crucial because radioactive decay follows an exponential pattern, often leading to quick initial decreases followed by slower rates of decay as the mass decreases.

Radioactive decay can occur naturally over time, but it can also be influenced by other factors, such as temperature or pressure. In this mathematical model, we are focusing solely on the time-dependent decay.

The initial value problem in differential equations involves solving an equation with given initial conditions. In this exercise, the initial condition is \( S(0) = 20 \), meaning at time \( t = 0 \), the mass of the substance is 20 grams.

This initial condition is crucial to determine the constant of integration when solving the differential equation. Without this information, we could only find a general solution, not a specific one that applies to this situation. It explains how the mass behaves starting from a known value, which forms the basis for predicting future behavior under given conditions.

Having the initial condition allows us to plug this value into our integrated equation to solve for the constant \( C_1 \). Incorporating this constant correctly ensures that the solution accurately reflects the system's behavior from the beginning.

Limiting behavior describes how a function behaves as the input approaches a certain value, often infinity. In this context, it's about understanding what happens to the mass \( S(t) \) as time \( t \) goes towards infinity.

The solution to the differential equation here shows \( S(t) = 10 - 10e^{-\frac{t}{10}} \). As \( t \rightarrow \infty \), the exponential term \( e^{-\frac{t}{10}} \) approaches zero. This means that the mass approaches a limit of 10 grams.

This is a form of equilibrium that a radioactive substance can reach despite ongoing processes of creation and decay. The limiting behavior helps us to predict long-term outcomes, highlighting that even with a constant creation of new substance, the decay process models out to a steady amount over time.

A first-order linear differential equation is one of the simplest forms of differential equations. It involves only the first derivative of the variable and has linear terms. The general form is \( \frac{dS}{dt} = aS + b \), where \( a \) and \( b \) are constants.

In this exercise, the equation \( \frac{dS}{dt} = 1 - \frac{1}{10}S \) is a first-order linear differential equation with \( a = -\frac{1}{10} \) and \( b = 1 \). This equation models processes where the rate of change of a quantity depends linearly on the amount present and constant external factors.

Solving these equations typically involves either separation of variables or integrating factors. Here, separation of variables allowed us to rearrange the terms and integrate them easily. Understanding first-order linear differential equations is important because they often appear in modeling real-world phenomena such as population growth, cooling processes, and of course, radioactive decay.