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

First order differential equations

What is a differential equation?

• A differential equation is simply an equation that contains one or more derivatives

  • For example $\frac{\mathrm{d}y}{\mathrm{d}x}=12x{y}^2$ is a differential equation

  • And so is $\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}-5\frac{\mathrm{d}x}{\mathrm{d}t}+7x=5\sin t$

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 $\frac{\mathrm{d}y}{\mathrm{d}x}=12x{y}^2$ is a first order differential equation

  • But $\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}-5\frac{\mathrm{d}x}{\mathrm{d}t}+7x=5\sin t$ is not a first order differential equation, because it contains the second derivative $\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}$

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

• Yes you have!

  • For example $\frac{\mathrm{d}y}{\mathrm{d}x}=3x^2$ 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