Ordinary Differential Equations/Basic Concepts

What is a differential equation?
A differential equation is any equation that has a derivative of a function. Examples of differential equations are The first example is the simplest differential equation with only a first derivative of the unknown function $$ y(x) $$ and nothing else. The other differential equations are more interesting and include the unknown function with and without a derivative, terms with the independent variable, terms multiplying the unknown function with its derivative, and more complex functions of the independent variable.
 * $$ y'(x) = 0 $$
 * $$ y'(x) + y^{2} = 0 $$
 * $$ \frac{du}{dx} = 4u + x^2 $$
 * $$ \frac{dx}{dt}x = \sin(t)e^{-t} $$

Ordinary Differential Equations (ODEs)
An ODE is a differential equation where the unknown function has one independent variable, like $$ y(x) $$, $$ x(t) $$ or $$ f(z) $$. Notation for ODEs can be The $$ y' $$, $$ y $$, $$ y' $$ notation, pronounced "y" prime, "y" double prime, "y" triple prime, is the most commonly used since it's compact to write, unlike the $$ \frac{dy}{dx} $$ fraction, with the $$ (x) $$ in $$ y'(x) $$ left off.
 * $$ \frac{dy}{dx}(x) = \frac{dy}{dx} = y'(x) = y' = y_{x}(x) = y_{x} = y^{(1)}(x) = y^{(1)}$$
 * $$ \frac{d^{2}y}{dx^{2}} = y'' = y^{(2)} $$
 * $$ \frac{d^{5}y}{dx^{5}} = y' = y^{(5)} $$
 * $$ \frac{du}{dt} = u'(t) = \dot{u} $$
 * $$ \frac{d^{2}u}{dt^{2}} = u''(t) = \ddot{u} $$

The $$ y_{x} $$ is another notation that is more common for PDEs and is discussed below.

When there are too many derivatives and counting the number of $$ ' $$ prime symbols become difficult, the $$y' = y^{(5)} $$ notation is used, with the parentheses meaning derivative rather than $$y^{5} = y \cdot y \cdot y \cdot y \cdot y $$ as $$ y $$ to the fifth power.

The dot notation of $$ u'(t) = \dot{u}$$ and $$ u''(t) = \ddot{u} $$, pronounced "u" dot and "u" double dot, is a special notation for derivatives with respect to time $$ t $$ and is common in engineering and physics.

Partial Differential Equations (PDEs)
A PDE is a differential equation where the unknown function has more than one independent variable, like $$ u(x,y,z) $$ or $$ f(x,t) $$. Examples are PDEs are much harder to solve in general than ODEs and have different methods. A good foundation in ODEs is necessary before trying PDEs, which is a course all on its own.
 * $$ \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} + \frac{\partial^{2} u}{\partial z^{2}} = 0 $$ (Laplace's equation)
 * $$ \frac{\partial f}{\partial t} + c\frac{\partial f}{\partial x} = 0 $$ (Advection equation)

Order of an ODE
The order of an ODE is the term with the highest derivative.
 * $$ y'(x) = 0 $$ is a first order ODE.
 * $$ y'' = x^{8} $$ is a second order ODE.
 * $$ y^{(5)} + y^{(4)} + y^{(3)} = 0 $$ is a fifth order ODE.
 * $$ y^{(2)} y^{(7)} = 0 $$ is a seventh order ODE.
 * $$ (y^{(4)})^3 = y^{(4)}y^{(4)}y^{(4)} = 0 $$ is a fourth order ODE.

Linear and Non-linear ODEs
A linear ODE is linear in the dependent variable. In other words, an ODE of $$ y(x) $$ can be written with all the $$ y(x)$$ terms and their derivatives not multiplying each other, to the first power, and not inside any functions. A non-linear ODE is any ODE that isn't linear.
 * $$ y' = 0 $$ is a linear ODE.
 * $$ y' = x^{8} $$ is a linear ODE.
 * $$ y'x^{8} = 1 $$ is a linear ODE.
 * $$ (y')^{2} = 0 $$ is a non-linear ODE.
 * $$ y'y = 0 $$ is a non-linear ODE.
 * $$ \sin(y') = 4xy $$ is a non-linear ODE.
 * $$ y'' + y = \ln(x) $$ is a linear ODE.
 * $$ y'' + y'y = \ln(x) $$ is a non-linear ODE.
 * $$ y''' + e^{y} = 7 $$ is a non-linear ODE.