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IBMathsDPAnalysis & Approaches (AA)HLRevision Notes5. CalculusDifferential EquationsIntroduction to First Order Differential Equations

Introduction to First Order Differential Equations (DP IB Analysis & Approaches (AA)): Revision Note

Written by: Roger B

Reviewed by: Dan Finlay

Updated on 15 July 2025

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First order differential equations

What is a differential equation?

A differential equation is simply an equation that contains one or more derivatives
- For example $\frac{d y}{d x} = 12 x y^{2}$ is a differential equation
- And so is $\frac{d^{2} x}{d t^{2}} - 5 \frac{d x}{d t} + 7 x = 5 \sin t$

What is a first order differential equation?

A first order differential equation is a differential equation that contains first derivatives but no second (or higher) derivatives
- For example $\frac{d y}{d x} = 12 x y^{2}$ is a first order differential equation
- But $\frac{d^{2} x}{d t^{2}} - 5 \frac{d x}{d t} + 7 x = 5 \sin t$ is not a first order differential equation, because it contains the second derivative $\frac{d^{2} x}{d t^{2}}$

Wait – haven’t I seen first order differential equations before?

Yes you have!
- For example $\frac{d y}{d x} = 3 x^{2}$ is also a first order differential equation, because it contains a first derivative and no second (or higher) derivatives
- But for that equation you can just integrate to find the solution y = x³ + c (where c is a constant of integration)
In this section of the course you learn how to solve differential equations that can’t just be solved right away by integrating

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Author: Roger B

Expertise: Maths Content Creator

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

Reviewer: Dan Finlay

Expertise: Maths Subject Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.