Differential Equations in Modelling

This lesson covers first-order differential equations with separable variables and their application to mathematical modelling. Differential equations are used to describe how quantities change in real-world situations and are a key part of the A-Level Mathematics specification.

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What Is a Differential Equation?

A differential equation is an equation involving derivatives. A first-order ordinary differential equation (ODE) involves dy/dx (or equivalent notation):

dy/dx = f(x, y)

The solution to a differential equation is a function y = g(x) that satisfies the equation.

• A general solution contains an arbitrary constant C.
• A particular solution has a specific value of C determined by an initial condition or boundary condition.

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Separable Variables

A first-order ODE is separable if it can be written in the form:

dy/dx = f(x) × g(y)

To solve, separate the variables — put all y terms on one side and all x terms on the other:

(1/g(y)) dy = f(x) dx

Then integrate both sides:

∫ (1/g(y)) dy = ∫ f(x) dx

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Example 1: Basic Separable Equation