Introduction to differential equations

Differential equations help us understand and predict behaviours in STEM, where they are used to model phenomena like motion, heat transfer, population dynamics, and financial markets. Use this resource to learn about first order separable differential equations.

Differential equations

A differential equation is an equation that includes a function and one or more of its derivatives. Differential equations shows how a quantity changes over time or space. They are used to model real-world situations where things are changing, like how fast a car is moving, how heat spreads through a metal, or how populations grow.

Consider the function \(y=f(x)\), where \(y'\) is the derivative of \(y\). Some examples of differential equations that involve this function are:

• \(y'=3x\)
• \(y'+2y=x+7\)
• \(y''-3y+8y=21e^{-8x}\)

Order of differential equations

The order of a differential equation is the highest order of any derivative of the unknown function \(y\).

If \(y'\) has an order of \(1\):

• \(y''\) has an order of \(2\)
• \(y'''\) has an order of \(3\)
• \(y^{4}\) has an order of \(4\)

...and so on.

The order of a differential equation helps us understand how complex it is to find a solution to the equation. If we can figure out the order, we can choose the right method to solve the differential equation.

Exercise – determining the order of differential equations

Find the order of the following differential equations.

1. \(y'=3x\)
2. \(y'+2y=x+7\)
3. \(y''-3y+8y=21e^{-8x}\)

Solving differential equations

To solve a differential equation, we look for a function or set of functions that satisfy the equation. This involves determining the function that, when subsituted into the differential equation, makes the equation true.

The method we use to solve the equation depends on the type of differential equation it is. We will cover: