Construct simple differential equations in pure mathematics and in context (contexts may include kinematics, population growth and modelling the relationship between price and demand)

Differential equations are equations that involve derivatives, representing rates of change. In A-Level Mathematics, you learn to construct simple differential equations from given information, both in pure mathematical contexts and real-world scenarios. This topic bridges algebraic manipulation with applied modelling, allowing you to describe how quantities evolve over time or space. Mastering this skill is essential for understanding more complex models in physics, biology, economics, and engineering.

In pure mathematics, you might derive a differential equation from a relationship involving a function and its derivative, such as dy/dx = ky. In context, you learn to translate word problems into mathematical equations. For example, in kinematics, acceleration is the derivative of velocity, leading to equations like dv/dt = a. In population growth, the rate of change of population is proportional to the current population, giving dP/dt = kP. Similarly, in economics, the relationship between price and demand can be modelled using differential equations where the rate of change of demand depends on price.

This topic is crucial because it teaches you to think like a mathematician: identifying variables, interpreting rates, and constructing equations that capture the essence of a problem. It also prepares you for further study in calculus and differential equations at university. In exams, you are expected to set up differential equations from descriptions, solve them using separation of variables or integrating factors, and interpret solutions in context.