# 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

• $y'(x)=0$ • $y'(x)+y^{2}=0$ • ${\frac {du}{dx}}=4u+x^{2}$ • ${\frac {dx}{dt}}x=\sin(t)e^{-t}$ 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.

## Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs)

### 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

• ${\frac {dy}{dx}}(x)={\frac {dy}{dx}}=y'(x)=y'=y_{x}(x)=y_{x}=y^{(1)}(x)=y^{(1)}$ • ${\frac {d^{2}y}{dx^{2}}}=y''=y^{(2)}$ • ${\frac {d^{5}y}{dx^{5}}}=y'''''=y^{(5)}$ • ${\frac {du}{dt}}=u'(t)={\dot {u}}$ • ${\frac {d^{2}u}{dt^{2}}}=u''(t)={\ddot {u}}$ The $y'$ , $y''$ , $y'''$ 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.

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

• ${\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}=0$ (Laplace's equation)
• ${\frac {\partial f}{\partial t}}+c{\frac {\partial f}{\partial x}}=0$ (Advection equation)

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.

## 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.