Interpreting Solutions of Differential Equations (Pure Mathematics 3 – 9709)

Core Concept

A differential equation used in modelling typically takes a form such as:

where $x$ represents some measurable quantity (e.g. temperature, population, mass of substance) and $t$ represents time (or another independent variable).

Step 1 — Solve: Separate variables and integrate to obtain the general solution, which contains an arbitrary constant $c$.

Step 2 — Apply initial/boundary conditions: Substitute given values to find $c$, producing the particular solution.

Step 3 — Interpret: Extract meaning from the particular solution. This may include:

• Finding the value of $x$ at a specific time $t$.
• Finding the time $t$ at which $x$ reaches a specified value.
• Determining the long-run (limiting) behaviour as $t \to \infty$.
• Commenting on whether the model is realistic (e.g. does $x$ remain positive? Does it grow without bound or level off?).
• Identifying any initial value ($x$ when $t = 0$).

Identifying Limiting Behaviour

A common interpretation task is to find the value that $x$ approaches as $t \to \infty$. This often arises when the solution contains an exponential term $e^{-kt}$ with $k > 0$, since $e^{-kt} \to 0$ as $t \to \infty$.

Checking Consistency with the Model

Examiners may ask you to verify that the solution is consistent with stated conditions (e.g. "show that $x$ is always positive") or to describe how the quantity changes over time (increasing, decreasing, approaching a limit).