Modelling with Differential Equations

• Rates of Change: In a mathematical model, the derivative $\frac{dy}{dt}$ represents the instantaneous rate at which a variable $y$ changes with respect to time $t$. This is the fundamental building block for describing dynamic systems like radioactive decay, population growth, or chemical reactions.

• Variable Identification: Successful modelling requires clearly defining the variables involved, such as $V$ for volume or $T$ for temperature, and identifying which is the independent variable (usually time $t$) and which is the dependent variable.

• Proportionality Constants: The symbol $\propto$ (proportional to) is replaced by an equals sign and a constant $k$ in equations. This constant $k$ characterizes the specific physical properties of the system being studied, such as a growth rate or a decay constant.

• Direct Proportionality: When a rate is directly proportional to a variable, we write $\frac{dy}{dt} = ky$. This leads to exponential growth if $k > 0$ or exponential decay if $k < 0$, where the speed of change is determined by the current amount present.

• Inverse Proportionality: If a rate is inversely proportional to a variable, the relationship is expressed as $\frac{dy}{dt} = \frac{k}{y}$. In this case, as the variable $y$ increases, the rate of change decreases, which is common in scenarios involving physical resistance or dilution.

• Composite Rates: Models often involve multiple factors acting simultaneously, such as an inflow and an outflow. These are combined by summing the individual rates, keeping careful track of positive (increase) and negative (decrease) signs.

Proportionality Comparison

• General vs. Particular Solutions: A general solution contains an arbitrary constant $c$ and represents a family of possible outcomes. A particular solution uses a boundary condition (like $y=5$ when $t=0$) to find a specific numerical value for $c$, resulting in a single model that fits the given data.

• The 'k' Confusion: Many learners forget that the proportionality constant $k$ is almost always defined as a positive value in a setup, and the direction of change (increase/decrease) should be handled by an explicit sign in the equation. Using a negative $k$ in a growth equation without explanation will lead to incorrect behavior.

• Separation of Variables Errors: When solving the formulated model, ensure that variables are grouped correctly with their respective differentials before integrating. Terms like $dy$ and $dt$ must always be in the numerator of their respective sides.

• Long-term Realism: Students often ignore whether a model makes sense as $t \to \infty$. A model for population growth $\frac{dP}{dt} = kP$ suggests infinite growth, which is rarely physically possible; recognizing these limitations is a sign of high-level analytical understanding.