91Ó°ÊÓ

First-order differential equations like the one in the exercise are equations that involve the first derivative of a function. In this case, the function is denoted by \( P(t) \), and the exercise gives us \( \frac{dP}{dt} = -3 P(t) \). This equation tells us how \( P(t) \) changes with respect to time \( t \). Essentially, the rate of change of \( P(t) \) is proportional to its current value, with a proportionality constant of \(-3\) in this case.
The negative sign in the equation signals that \( P(t) \) decreases over time, particularly at a rate three times its current value. Understanding these basic elements of a first-order differential equation helps in recognizing how these types of equations describe scenarios where a quantity diminishes or escalates based on its current state.

Separation of variables is a method used to solve differential equations, particularly useful for first-order equations. The main idea is to manipulate the equation so that all terms involving one variable (here, \( P \)) are on one side, and all terms involving the other variable (here, \( t \)) are on the other side.
In the exercise, we rearrange \( \frac{dP}{dt} = -3 P(t) \) to \( \frac{dP}{P} = -3 \, dt \). This process sets the stage for integration, making it possible to find solutions by integrating both sides separately.

• By integrating \( \frac{dP}{P} \), we obtain \( \ln |P| \).
• By integrating \(-3 \, dt \), results in \(-3t + C\), where \( C \) is the constant of integration.

After integration, solving the resulting equation leads us to the general form of the solution, wherein the constant \( C \) can be adjusted according to initial or boundary conditions.

The concept of exponential decay is illustrated perfectly by the resolved form of this differential equation: \( P(t) = Ce^{-3t} \). Exponential decay occurs when the value of a quantity decreases at a rate proportional to its current value.
The formula \( P(t) = Ce^{-3t} \) models how \( P \) decreases exponentially over time. Here, \( e^{-3t} \) represents the decay factor that shrinks the function value, indicating a rapid decline as time progresses since the exponent \(-3t\) increases negatively with higher \( t \) value.

• The constant \( C \) alters the initial state or size of \( P(t) \). Larger initial values imply a larger \( C \).

Understanding exponential decay is crucial in fields like physics, biology, and finance, where processes change exponentially over time.