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.