Notation and Definitions | Topic Playlist

A differential equation is an equation that has an unknown function and its derivatives present. The simplest example is $y'(x) = y(x)$ with $y(x)$ an unknown function.

A non-example is $y'(x) = x$ which is not usually considered a differential equation since $y(x)$ is not present.

An ODE is a differential equation where the unknown function is a function of a single variable, like $y(x)$.

A PDE is a differential equation where the unknown function is a function of a multiple variables, like $f(x,y)$ or $f(x,y,z)$. An example is the temperature at every point $(x,y)$ on a flat metal plate or the density of air at every point $(x,y,z)$ in a music hall.

There's three common ways to write derivatives of a single variable function $y(x)$. Which one to use is personal preference, but people usually pick the one that requires the least writing and is easiest to read.

\begin{equation} \frac{\text{d}y}{\text{d}x} = y' = y^{(1)} \qquad\qquad \frac{\text{d}^{2}y}{\text{d}x^{2}} = y'' = y^{(2)} \qquad\qquad \frac{\text{d}^{5}y}{\text{d}x^{5}} = y''''' = y^{(5)} \end{equation}

A fourth way to write derivatives of a single variable function $u(t)$ is with the "dot notation" common in physics and engineering.

\begin{equation} \frac{\text{d}u}{\text{d}t} = u' = u^{(1)} = \dot{u} \qquad\qquad \frac{\text{d}^{2}u}{\text{d}t^{2}} = u'' = u^{(2)} = \ddot{u} \end{equation}

This notation is used for functions of time $t$ and you typically only see a single "u dot" or a "u double dot" since first and second time derivatives are common in physics and engineering. A differential equation with a "u triple dot" is uncommon.

There's two common ways to write partial derivatives for a function $f(x,y)$. (Note: The order on the mixed partial changes between the two notations.)

\begin{equation} \frac{\partial f}{\partial x} = f_{x} \qquad \qquad \frac{\partial^{2} f}{\partial x^{2}} = f_{xx} \qquad\qquad \frac{\partial f}{\partial y} = f_{y} \qquad \qquad \frac{\partial^{2} f}{\partial y^{2}} = f_{yy} \end{equation} \begin{equation} \frac{\partial }{\partial y}\frac{\partial }{\partial x} f = \frac{\partial^{2} f}{\partial y \partial x} = f_{xy} \qquad\qquad \frac{\partial }{\partial x}\frac{\partial }{\partial y} f = \frac{\partial^{2} f}{\partial x \partial y} = f_{yx} \end{equation}

The order of a differential equation is the highest derivative of the unknown function present.

\begin{align} y'' + y = 1 & \quad\text{2nd order} \\ \\ (y')^{5} + y = 1 & \quad\text{1st order} \\ \\ y'''y''y'y = x & \quad\text{3rd order} \\ \\ \sin(y^{(4)}) + y = e^{x} & \quad\text{4th order} \\ \\ \end{align}

A linear differential equation has the following form.

\begin{equation} p_{n}(x)y^{(n)} + p_{n-1}(x)y^{(n-1)} + \cdots + p_{1}(x)y' + p_{-}(x)y = g(x) \end{equation}

The functions of $x$ do not matter for the differential equation to be linear or nonlinear. The only thing that matters is the $y(x)$ function and all its derivatives multiply a function of $x$. A differential equation is nonlinear means the differential equation does not fit the form of a linear differential equation.

\begin{align} y'' + y = 1 & \quad\text{linear} \\ \\ \tan\left(\frac{\ln(x)}{x^{2}+1}\right) y'' + x^{x} y = e^{\sin(x)} & \quad\text{linear} \\ \\ (y')^{2} + y = 7 & \quad\text{nonlinear} \\ \\ y''y' + y' + xy = 31 & \quad\text{nonlinear} \\ \\ y^{(5)} + \frac{1}{y} = e^{x} & \quad\text{nonlinear} \\ \\ y'' + \sin(y') + y = \frac{1}{x} & \quad\text{nonlinear} \\ \\ y'' + e^{y} = 4 & \quad\text{nonlinear} \\ \\ \ln(y') + y = 0 & \quad\text{nonlinear} \\ \end{align}

An explicit solution to an ODE is $y(x)=f(x)$. Ideally, we want to solve for $y$ in terms of $x$, but that is not always possible.

For example, there is no way to solve for $y$ in the following equation.

\begin{equation} y^{2} + e^{y} = \sin(x) + C\ln(x) \end{equation}

This is an implicit general solution for $x$ and $y$.