# 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

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

The first example is the simplest differential equation with only a first derivative of the unknown function ${\displaystyle y(x)}$ 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 ${\displaystyle y(x)}$, ${\displaystyle x(t)}$ or ${\displaystyle f(z)}$. Notation for ODEs can be

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

The ${\displaystyle y'}$, ${\displaystyle y''}$, ${\displaystyle y'''}$ notation, pronounced "y" prime, "y" double prime, "y" triple prime, is the most commonly used since it's compact to write, unlike the ${\displaystyle {\frac {dy}{dx}}}$ fraction, with the ${\displaystyle (x)}$ in ${\displaystyle y'(x)}$ left off.

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

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

The dot notation of ${\displaystyle u'(t)={\dot {u}}}$ and ${\displaystyle u''(t)={\ddot {u}}}$, pronounced "u" dot and "u" double dot, is a special notation for derivatives with respect to time ${\displaystyle t}$ 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 ${\displaystyle u(x,y,z)}$ or ${\displaystyle f(x,t)}$. Examples are

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

• ${\displaystyle y'(x)=0}$ is a first order ODE.
• ${\displaystyle y''=x^{8}}$ is a second order ODE.
• ${\displaystyle y^{(5)}+y^{(4)}+y^{(3)}=0}$ is a fifth order ODE.
• ${\displaystyle y^{(2)}y^{(7)}=0}$ is a seventh order ODE.
• ${\displaystyle (y^{(4)})^{3}=y^{(4)}y^{(4)}y^{(4)}=0}$ 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 ${\displaystyle y(x)}$ can be written with all the ${\displaystyle y(x)}$ 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.

• ${\displaystyle y'=0}$ is a linear ODE.
• ${\displaystyle y'=x^{8}}$ is a linear ODE.
• ${\displaystyle y'x^{8}=1}$ is a linear ODE.
• ${\displaystyle (y')^{2}=0}$ is a non-linear ODE.
• ${\displaystyle y'y=0}$ is a non-linear ODE.
• ${\displaystyle \sin(y')=4xy}$ is a non-linear ODE.
• ${\displaystyle y''+y=\ln(x)}$ is a linear ODE.
• ${\displaystyle y''+y'y=\ln(x)}$ is a non-linear ODE.
• ${\displaystyle y'''+e^{y}=7}$ is a non-linear ODE.