7_differential_equations_1.md

title: Differential Equations

Differential equations 1

The first lecture on differential equations consists of three parts, each with their own video:

• First examples of differential equations
• Theory of systems of first-order differential equations
• Solving homogeneous first-order differential equations with constant coefficients

Total video length: 1 hour 15 minutes 4 seconds

First examples of differential equations: Definitions and strategies

Definitions

A differential equation or DE is any equation which involves both a function and some derivative of that function. In this lecture we will be focusing on Ordinary Differential Equations (ODEs), meaning that our equations will involve functions of one independent variable and hence any derivatives will be full derivatives. Equations which involve a function of several independent variables and their partial derivatives are called Partial Differential Equations (PDEs). and will be introduced in the follow up lecture.

We consider functions x(t) and define \dot{x}(t)=\frac{dx}{dt}, x^{(n)}(t)=\frac{d^{n}x}{dt^{n}}. An n-th order differential equation is an equation of the form 5

x^{(n)}(t) = f(x^{(n-1)}(t), \cdots, x(t), t).