1.1. Definitions, Classification and Terminology#
Definition 1.1 (Differential Equations)
An equation containing the derivatives of one or more unknown functions (or dependent variables), with respect to one or more independent variables, is said to be a differential equation (DE).
We can classify differential equations by type, order, linearity and homogeneity.
Classification by Type
Ordinary Differential Equation (ODE)
If a differential equation contains only ordinary derivatives of one or more unknown functions with respect to a single independent variable, it is said to be an ordinary differential equation (ODE).
Example 1.1
The following equations are ordinary differential equations:
\[ \diff{y}{x} + 5y = e^x, \quad \diff[2]{y}{x} - \diff{y}{x} + 6y = 0, \quad \diff{x}{t} + \diff{y}{t} = 2x + y \]In the first two equations, \(y=y(x)\).
In the third equation, \(x=x(t)\) and \(y=y(t)\).
The functions here are functions of a single variable.
Partial Differential Equation (PDE)
An equation involving partial derivatives of one or more unknown functions of two or more independent variables is called a partial differential equation (PDE).
Example 1.2
The following equations are partial differential equations:
\[ \pdiff[2]{u}{x} + \pdiff[2]{u}{y} = 0, \quad \pdiff[2]{u}{x} = \pdiff[2]{u}{t} - 2\pdiff{u}{t}, \quad \pdiff{u}{y}=-\pdiff{v}{x} \]In the first equation, \(u=u(x, y)\).
In the second equation, \(u=u(x, t)\).
In the second equation, \(u=u(y,\ldots)\), \(v=v(x,\ldots)\)
The functions here are multi-variable functions.
Remark 1.1 (Notations of derivatives)
Ordinary derivatives can be written by using
Leibniz notation
\(\displaystyle \diff{y}{x}\), \(\displaystyle \diff[2]{y}{x}\), \(\displaystyle \diff[3]{y}{x}\), …
Prime notation
\(y'\), \(y''\), \(y'''\), \(y^{(4)}\), …, \(y^{(k)}\)
Newton’s dot notation
\(\dot y\), \(\ddot y\)
Classification by Order
The order of a differential equation is the order of the highest derivative in the equation.
Example 1.3
The following equation
\[ \diff[2]{y}{x} + 5 \left(\diff{y}{x}\right)^3 - 4y = e^x \]is a second-order ordinary differential equation.
Remark 1.2 (Order and Degree of differential equations)
Order: The order of a differential equation is defined to be that of the highest order derivative it contains.
Degree: The degree of a differential equation is defined as the power to which the highest order derivative is raised.
The equation
\[ \left(y'''\right)^2 + \left(y''\right)^4+ y =x \]is a second-degree, third-order differential equation.
Remark 1.3 (Forms of Differential Equations)
For an \(n\)-th order ordinary differential equation, we can express it in
General Form
\[ F(x, y, y', y'', \ldots, y^{(n)}) = 0 \]
Normal form
\[ \diff[n]{y}{x} = f(x, y', y'', \ldots, y^{(n-1)}) \]For example:
A first-order ordinary differential equation: \(\displaystyle \diff{y}{x}= f(x, y)\)
A second-order ordinary differential equation: \(\displaystyle \diff[2]{y}{x}= f(x, y, y')\)
Classification by Linearity
Linear
An \(n\)-th order ordinary differential equation is linear if it can be expressed as
(1.1)#\[ a_n(x) y^{(n)} + a_{n-1}(x) y^{(n-1)} + \cdots + a_1(x) y' + a_0(x) y = g(x), \quad a_n(x)\neq 0 \]with two properties
The dependant variable \(y\) and all its derivatives \(y'\), \(y''\), … \(y^{(n)}\) are of the first degree.
The coefficients \(a_0(x) \), \(a_1(x) \), …, \(a_n(x) \) depend at most on the independent variable \(x\).
Nonlinear
A non-linear ordinary differential equation is one that is not linear. If an ordinary differential equation involves any of the followings
\(y^2\), \((y')^2\)
\(\sin y\), \(\sin y'\), \(\cos y\), \(\cos y'\)
\(e^y\), \(e^{y'}\)
it is then non-linear.
Classification by Homogeneity
Homogeneous
Equation (1.1) is homogeneous if \(g(x) \equiv 0\).
Nonhomogeneous
Equation (1.1) is nonhomogeneous if \(g(x)\neq 0\).