The Grammar of Dynamics
A differential equation is an equation that contains an unknown function and some of its derivatives. To speak the language of DEs, we must identify the roles of variables:
- Independent Variable ($t$): Typically represents time or position.
- Dependent Variable ($P$ or $y$): Represents the state of the system (e.g., population size).
- Order: The highest derivative present in the equation. For example, $y'' + y = 0$ is a second-order equation.
The Model of Natural Growth
Consider the law of natural growth: the rate of change of a population is directly proportional to its size. This translates to the first-order DE:
$$\frac{dP}{dt} = kP$$
Here, $k$ is the relative growth rate. This model suggests that the larger the population, the faster it grows—a hallmark of exponential behavior.
Verifying Solutions
How do we know if a function is a solution? It must satisfy the identity for all $t$.
Let $P(t) = Ce^{kt}$. We compute the derivative:
$$P'(t) = \frac{d}{dt}(Ce^{kt}) = C(ke^{kt}) = k(Ce^{kt})$$
Since $Ce^{kt} = P(t)$, we have $P'(t) = kP(t)$. The identity holds!
Initial Conditions and Uniqueness
The solution $P = Ce^{kt}$ is actually a family of solutions. To find a specific curve, we need an initial condition, such as $P(0) = P_0$. This physical constraint allows us to solve for $C$, identifying the unique trajectory of our system. Note: In biological contexts, we restrict $C > 0$ because populations cannot be negative.