book:content:defs - The Geometry of Linear Algebra

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\subsection{Definitions}
%*/
%*=== Definitions ===

\begin{enumerate}\item
A \textbf{differential equation} is an equation involving an
unknown function and its derivatives.

\item  A differential equation is an \textbf{ ordinary} differential
equation if the unknown function depends on only one independent
variable, otherwise it is a \textbf{partial} differential equation.

\item  The \textbf{order} of a differential equation is the order
(number of derivatives taken) of the highest derivative appearing in
the equation.

\item  The \textbf{degree} of a differential equation that can be
written as a polynomial in the unknown function and its derivatives is
the power to which the highest order derivative is raised.

\item  An $n^{th}$ order differential equation is \textbf{linear} if
it has the form

\begin{equation}
a_n(x) y^{(n)}(x) + a_{n-1}(x) y^{(n-1)}(x) + \dots + a_0(x) y(x) = b(x)\label{linearinhomo}
\end{equation}

A linear equation is \textbf{homogeneous} if $b(x) = 0$ and \textbf{inhomogeneous} if $b(x)\ne 0$.

\item  A \textbf{solution} of a differential equation in the unknown
function $y$ and the independent variable $x$ on the interval $I$ is a
function $y(x)$ that satisfies the differential equation identically
for all $x$ in $I$.

\item  A \textbf{particular} solution is any one solution.  The \textbf{general solution} is the set of all solutions.

\item  An \textbf{initial value problem} is a differential equation
together with conditions (known as initial conditions) on the unknown
function and its derivatives all at the same value of the independent variable.

\item  A \textbf{boundary value problem} is a differential equation
together with conditions (known as boundary conditions) on the unknown
function and its derivatives at more than one value of the independent
variable. 
\end{enumerate}

%/*
\subsection{Notation}
%*/
%*=== Notation ===

There are several different notations for derivatives in common use.  You should be comfortable with all of them.
Leibniz's notation for derivatives is: 
\begin{equation}
\frac{dy}{dx}, \qquad \frac{d^2 y}{dx^2}\qquad\frac{d^3 y}{dx^3}\qquad\dots\qquad\frac{d^n y}{dx^n}
\end{equation}
Lagrange's notation for the same derivatives (wrt to the independent variable $x$) is:
\begin{equation}
y^{\prime} \qquad y^{\prime\prime}\qquad y^{\prime\prime\prime} \qquad \dots \qquad y^{(n)}
\end{equation}
Newton, who used time ($t$) as the independent variable, rather than $x$, used \textit{dots} instead of \textit{primes}:
\begin{equation}
\dot{y} \qquad \ddot{y} \qquad \dots \qquad y^{(n)}
\end{equation}

The equation for the general linear inhomogeneous differential equation, eqn.(\ref{linearinhomo}) above is long and messy looking.  To simplify this equation, it is common to pull out all the derivative operators with their coefficients into a single \textit{differential operator} acting on the unknown function $y$ and denoting the big messy operator by the single caligraphic letter $\cal{L}$, i.e.
$$\cal{L}\equiv a_n(x) \frac{d^{n}}{dx^{n}} + a_{n-1}(x) \frac{d^{n-1}}{dx^{n-1}} + \dots + a_0(x)$$
so that eqn.(\ref{linearinhomo}) becomes
$$\cal{L} y(x)=b(x).$$
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