Definition of a differential equation

A differential equation is one that relates the independent variable \(x\), the dependent variable \(y\) as a function of \(x\) and its derivative with respect to the independent variable \(x\).

Examples of differential equations are:

1. \[ \frac{\mathrm{d}y}{\mathrm{d}x} = \cos x. \]

2. \[ x^2 \frac{\mathrm{d}y}{\mathrm{d}x} = y^2 + 2. \]

3. \[ \frac{\mathrm{d}^2y}{\mathrm{d}x^2} = 6x - 2. \]

A first order differential equation involves only the first derivative of the dependent variable with respect to the independent one.
Examples (i) and (ii) above are examples of first order differential equations.

A solution of a differential equation is a function \(y(x)\) -- expressing an explicit relation (through an equation) between the independent and dependent variables -- which satisfies the differential equation for all values of \(x\) in the domain.

In this topic, we will be exploring different methods of solving first order differential equations.