Differential Equations

This lesson covers first-order differential equations solved by separating variables — as required by the Edexcel A-Level Mathematics specification (9MA0). You need to be able to separate variables, integrate both sides, find general and particular solutions, and form differential equations from contextual problems.

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What Is a Differential Equation?

A differential equation is an equation that involves derivatives (rates of change). For example:

• dy/dx = 3x²
• dy/dx = 2y
• dy/dx = xy + 1

At A-Level, you focus on first-order differential equations (involving dy/dx only, not d²y/dx²).

General Solution vs Particular Solution

• The general solution contains the constant of integration c and represents a family of curves.
• A particular solution is found by using a boundary condition (an initial condition or known point) to determine c. It represents one specific curve from the family.

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Separating Variables

The method of separation of variables works when the differential equation can be written in the form:

dy/dx = f(x) × g(y)

That is, the right-hand side can be expressed as a product of a function of x only and a function of y only.

Step-by-Step Method