Definitions

Section 15.1 Definitions

In this section, we define several different types of differential equations and discuss their properties.

Definition 15.1. Differential Equations.

A differential equation is an equation involving an unknown function and its derivatives. A differential equation is an ordinary differential equation if the unknown function depends on only one independent variable, otherwise it is a partial differential equation. The order of a differential equation is the order (number of derivatives taken) of the highest derivative appearing in the equation. The 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.

A simple example of a second-order ordinary differential equation is

\begin{equation} \frac{d^2y}{dx^2} = a\tag{15.1.1} \end{equation}

where \(a\) is a constant. This simple example can be solved by integrating both sides twice.

Definition 15.2. Linearity and Homogeneity.

An \(n^{\hbox{th}}\) order differential equation is 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) .\tag{15.1.2} \end{equation}

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

The example above is inhomogeneous unless \(a=0\text{,}\) in which case it is homogeneous.

Notation 15.1. Linear Differential Operators.

To simplify differential equations such as (15.1.2), it is common to pull out all the derivative operators with their coefficients into a single differential operator acting on the unknown function \(y\text{.}\) Thus, the big messy operator gets replaced by the single caligraphic letter \(\LL\text{,}\) where

\begin{equation} \LL\equiv a_n(x) \frac{d^{n}}{dx^{n}} + a_{n-1}(x) \frac{d^{n-1}}{dx^{n-1}} + \dots + a_0(x)\tag{15.1.3} \end{equation}

so that (15.1.2) becomes

\begin{equation} \LL y(x)=b(x)\text{.}\tag{15.1.4} \end{equation}

By convention, the differential operator \(\LL\) is always written to the left of the function that it acts on.

In the example above, \(\LL=\frac{d^2}{dx^2}\text{.}\)

Definition 15.3. Solution (of a differential equation).

A 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 \(I\text{.}\) A particular solution of a differential equation is any one solution. The general solution is the set of all solutions, typically expressed in terms of free parameters.

The general solution of the example differential equation is \(y=\frac{a}2 x^2+bx+c\text{,}\) where \(b\) and \(c\) are free parameters, that is, constants that can take on any value.

Definition 15.4. Initial and Boundary Value Problems.

An 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. A 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.

The constants \(b\) and \(c\) could be determined by specifying either the initial values \(y(0)\) and \(y'(0)\) or the boundary values \(y(x_1)\) and \(y(x_2)\text{.}\)