```{index} Complex Number
```
```{index} Complex Number; Complex Conjugate
```

(chap01:DE)=
# Simple Differential Equations and Applications

````{prf:definition} Differential Equation
A differential equation that involves only one independent variable is called
an ordinary differential equation (ODE). Those involving two or more independent variables are called partial differential equations (PDEs). This chapter will be concerned with ODEs only.
````

The subject of ODEs encompasses analytical, computational, and applicable fields of interest. There are many textbooks written from the elementary to the most advanced, with some focusing on applications and others concentrating on existence theorems and rigorous methods of solution. This chapter is intended to introduce the reader to all three branches of the subject. For more information the reader should consult the ODE textbooks [1] – [8].

## Separable Differential Equations

Consider the differential equation

$$
\diff{x}{t}=f(t,x)
$$(eq:ode:separable)

## Exact Differential Equations


## Homogeneous First-order Differential Equations

## Linear Differential Equations

## Poincare Maps