Preliminaries

Notation

In this course we will use three different notations to denote derivatives:

Leibniz’s notation:

\[ \frac{\mathrm{d}y}{\mathrm{d}x}, \frac{\mathrm{d}^2y}{\mathrm{d}x^2},\ldots\frac{\mathrm{d}^n y}{\mathrm{d}x^n}. \]

Newton’s notation (dots usually are used for derivatives w.r.t. time):

\[ \dot{x}\equiv\frac{\mathrm{d}x}{\mathrm{d}t},\quad\ddot{x}\equiv\frac{\mathrm{d}^2y}{\mathrm{d}x^2},\ldots \]

Lagrange’s notation:

\[ y'(x)\equiv\frac{\mathrm{d}y}{\mathrm{d}x},\quad y''(x)\equiv\frac{\mathrm{d}^2y}{\mathrm{d}x^2}, \; \ldots,\quad y^{(n)}(x)\equiv\frac{\mathrm{d}^ny}{\mathrm{d}x^n}. \]

Order and degree

The order of the derivative \(\frac{\mathrm{d}^ny}{\mathrm{d}x^n}\) is \(n\).

The order of a differential equation is the order of the highest order derivative that appears in the equation.

The degree of a differential equation is the power to which the highest order derivative is raised.

Example

The ordinary differential equation (ODE)

\[ \left(\frac{\mathrm{d}^5y}{\mathrm{d}x^5}\right)^3-2x\frac{\mathrm{d}y}{\mathrm{d}x}=2xy \]

is 5th order (because the highest order derivative is a 5th order derivative) and 3rd degree (because the highest order [i.e. 5th order] derivative is raised to the power 3).

Linearity

Linearity is a very important concept in mathematics which you will meet in many mathematics modules. You’ll meet the most general definition in Linear Algebra next year, but in terms of ODEs the definition can be taken to be as follows:

An ODE of order \(n\) is linear if it can be written in the form

\[ a_n(x)\frac{\mathrm{d}^ny}{\mathrm{d}x^n}+a_{n-1}(x)\frac{\mathrm{d}^{n-1}y}{\mathrm{d}x^{n-1}}+\ldots+a_1(x)\frac{\mathrm{d}y}{\mathrm{d}x}+a_0(x)y+q_0(x)=0. \]

Here, \(a_n(x),a_{n-1}(x),\ldots a_{0}(x)\) and \(q_0(x)\) are functions of the independent variable \(x\) only - i.e. they do not depend on the dependent variable \(y\). Note that any of these functions may be zero or constant - these are just special cases of functions of \(x\).

Examples

• The third order ODE

  \[ 3\frac{\mathrm{d}^3y}{\mathrm{d}x^3}+2x\frac{\mathrm{d}y}{\mathrm{d}x}+\cos(x)y +\sin(x)=0 \]

  is linear, because it can be written in the form in the definition with

  \[ a_3(x)=3,\quad a_2(x)=0, \quad a_1(x)=2x,\quad a_0(x)=\cos(x),\quad q_0(x)=\sin(x). \]

• The second order ODE

  \[ xy\frac{\mathrm{d}y}{\mathrm{d}x}=\sin(x) \]

  is non-linear, because the co-efficient of the \(\frac{\mathrm{d}y}{\mathrm{d}x}\) term depends on \(y\).

• The third order ODE

  \[ 4\frac{\mathrm{d}^3y}{\mathrm{d}y^3}+\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^3=0 \]

  is non-linear because the term \(\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^3\) cannot be written as the product of \(\frac{\mathrm{d}y}{\mathrm{d}x}\) with a function which does not involve \(y\).