Definition:Differential Equation
Definition
A differential equation is a mathematical equation for an unknown function of one or several variables relating:
- $(1): \quad$ The values of the function itself
- $(2): \quad$ Its derivatives of various orders.
Order
The order of a differential equation is defined as being the order of the highest order derivative that is present in the equation.
Degree
Let $f$ be a differential equation which can be expressed as a polynomial in all the derivatives involved.
The degree of $f$ is defined as being the power to which the derivative of the highest order is raised.
By default, if not specifically mentioned, the degree of a differential equation is assumed to be $1$.
Ordinary, Partial and Total Differential Equations
There are three types of differential equation:
Ordinary Differential Equation
An ordinary differential equation (abbreviated O.D.E. or ODE) is a differential equation which has exactly one independent variable.
All the derivatives occurring in it are therefore ordinary.
The general ODE of order $n$ is:
- $\map f {x, y, \dfrac {\d x} {\d y}, \dfrac {\d^2 x} {\d y^2}, \ldots, \dfrac {\d^n x} {\d y^n} } = 0$
or, using the prime notation:
- $\map f {x, y, y', y'', \ldots, y^{\paren n} } = 0$
Partial Differential Equation
A partial differential equation (abbreviated P.D.E. or PDE) is a differential equation which has:
- one dependent variable
- more than one independent variable.
The derivatives occurring in it are therefore partial.
Total Differential Equation
A total differential equation is a differential equation which contains:
- more than one dependent variable
- one independent variable which may or may not appear explicitly in that differential equation.
Linear and Non-Linear
Differential equations can also be classified as to whether they are linear or non-linear.
Linear
A linear differential equation is a differential equation where all dependent variables and their derivatives appear to the first power.
Neither are products of dependent variables allowed.
Non-Linear
A non-linear differential equation is a differential equation which is not linear.
Distributional
A differential equation is classified as distributional if, in addition to ordinary and partial derivatives and functions, they also involve at least one of the following notions:
- Distributional ordinary derivative
- Distributional partial derivative
- Distribution and/or its derivatives
Note that every standard differential equation can be written as a distributional one, but not the other way around.
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Solution
Let $\Phi$ be a differential equation defined on a domain $D$.
Let $\phi$ be a function which satisfies $\Phi$ on the whole of $D$.
Then $\phi$ is known as a solution of $\Phi$.
Note that, in general, there may be more than one solution to a given differential equation.
On the other hand, there may be none at all.
General Solution
Let $\Phi$ be a differential equation.
The general solution of $\Phi$ is the set of all functions $\phi$ that satisfy $\Phi$.
Particular Solution
Let $S$ denote the solution set of $\Phi$.
A particular solution of $\Phi$ is the element of $S$, or subset of $S$, which satisfies a particular boundary condition of $\Phi$.
Weak Solution
A weak solution is a solution of a non-standard formulation of a differential equation.
Autonomous
A differential equation is autonomous if none of the derivatives depend on the independent variable.
The $n$th order autonomous differential equation takes the form:
- $y^{\paren n} = \map f {y, y', y'', \dots, y^{\paren {n - 1} } }$
System of Differential Equations
A system of differential equations is a set of simultaneous differential equations.
The solutions for each of the differential equations are in general expected to be consistent.
Explicit and Implicit
Explicit System
A differential equation is called explicit if and only if it can be written in the form:
- $y^{\paren n} = \map f {x, y, y', y'', \dots, y^{\paren {n - 1} } }$
Implicit
A differential equation that is not explicit is referred to as implicit.
Examples
Second Order Linear Ordinary Differential Equation
Ordinary linear differential equation of the $2$nd order:
- $\dfrac {\d^2 y} {\d x^2} + y = x^2$
First Order First Degree Non-Linear Ordinary Differential Equation $(1)$
Ordinary non-linear differential equation of the $1$st order and $1$st degree:
- $\paren {x + y}^2 \dfrac {\d y} {\d x} = 1$
First Order First Degree Non-Linear Ordinary Differential Equation $(2)$
Ordinary non-linear differential equation of the $1$st order and $1$st degree:
- $\dfrac {\d y} {\d x} = \dfrac x {y^{1/2} \paren {1 + x^{1/2} } }$
First Order First Degree Non-Linear Ordinary Differential Equation $(3)$
Ordinary non-linear differential equation of the $1$st order and $1$st degree:
- $\dfrac {\d y} {\d x} = 1 + x y^2$
First Order Second Degree Non-Linear Ordinary Differential Equation
Differential Equation/Examples/First Order Second Degree Non-Linear Ordinary
Second Order Non-Linear Ordinary Differential Equation
Ordinary non-linear differential equation of the $2$nd order:
- $\dfrac {\d^2 y} {\d x^2} + \paren {3 \dfrac {\d y} {\d x} }^3 + 2 x = 7$
Second Order Second Degree Non-Linear Ordinary Differential Equation
Ordinary non-linear differential equation of the $2$nd order and $2$nd degree:
- $\paren {1 + \paren {\dfrac {\d y} {\d x} }^2}^{3/2} = 3 \dfrac {\d^2 y} {\d x^2}$
Third Order Linear Ordinary Differential Equation
Ordinary linear differential equation of the $3$rd order:
- $2 \dfrac {\d^3 y} {\d x^3} + 3 \dfrac {\d^2 y} {\d x^2} + \dfrac {\d y} {\d x} - 10 y = e^{-3 x} \sin 5 x$
Third Order Second Degree Non-Linear Ordinary Differential Equation
Ordinary non-linear differential equation of the $3$rd order and $2$nd degree:
- $\paren {\dfrac {\d^3 y} {\d x^3} }^2 + \paren {\dfrac {\d^2 y} {\d x^2} }^4 + \dfrac {\d y} {\d x} = x$
Fourth Order First Degree Non-Linear Ordinary Differential Equation
Ordinary non-linear differential equation of the $4$th order and $1$st degree:
- $x \dfrac {\d^4 y} {\d x^4} + 2 \dfrac {\d^2 y} {\d x^2} + \paren {x \dfrac {\d y} {\d x} }^5 = x^3$
First Order Linear Partial Differential Equation
Partial linear differential equation of the $1$st order in $2$ independent variables:
- $x \dfrac {\partial z} {\partial x} + y \dfrac {\partial z} {\partial y} - z = 0$
Second Order Linear Partial Differential Equation $(1)$
Partial linear differential equation of the $2$nd order in $3$ independent variables:
- $\dfrac {\partial^2 V} {\partial x^2} + \dfrac {\partial^2 V} {\partial y^2} + \dfrac {\partial^2 V} {\partial z^2} = 0$
Second Order Linear Partial Differential Equation $(2)$
Partial linear differential equation of the $2$nd order in $2$ independent variables:
- $\dfrac {\partial^2 y} {\partial t^2} = \alpha^2 \dfrac {\partial^2 y} {\partial x^2}$
Second Order Second Degree Non-Linear Partial Differential Equation
Partial non-linear differential equation of the $2$nd order and $2$nd degree in $2$ independent variables:
- $\dfrac {\partial^2 z} {\partial x^2} \cdot \dfrac {\partial^2 z} {\partial y^2} - \paren {\dfrac {\partial^2 x} {\partial x \partial y} }^2 = 0$
First Order First Degree Total Differential Equation
Total differential equation of the $1$st order and $1$st degree:
- $u \rd x + v \rd y + w \rd z = 0$
First Order Second Degree Total Differential Equation
Total differential equation of the $1$st order and $2$nd degree:
- $x^2 \rd x^2 + 2 x y \rd x \rd y + y^2 \rd y^2 - z^2 \rd z^2 = 0$
Also see
- Results about differential equations can be found here.
Historical Note
According to H.T.H. Piaggio, the first person to solve a differential equation was Isaac Newton, which he did in $1676$ by use of an infinite series, $11$ years after he had invented the differential calculus in $1665$.
These results were not published till $1693$, the same year in which a differential equation occurred in the work of Gottfried Wilhelm von Leibniz, whose own work on differential calculus was published in $1684$.
However, E.L. Ince states that the term differential equation was first used by Gottfried Wilhelm von Leibniz (as æquatio differentialis) also in $1676$, to denote a relationship between the differentials $\d x$ and $\d y$ of two variables $x$ and $y$.
Jacob Bernoulli and Johann Bernoulli reduced a large number of differential equations into forms that could be solved.
Much of the theory of differential equations was established by Leonhard Paul Euler.
Joseph Louis Lagrange gave a geometrical interpretation in $1774$.
The first existence proof for the solutions of a differential equation was provided by Augustin Louis Cauchy.
He proved in $1823$ that the infinite series obtained from a differential equation is convergent.
The theory in its present form was not presented until the work of Arthur Cayley in $1872$.
Piaggio references the $1888$ work of Micaiah John Muller Hill.
Cauchy's work was continued by Charles Auguste Briot and Jean-Claude Bouquet
The Method of Successive Approximations was introduced by Charles Émile Picard in $1890$.
Lazarus Immanuel Fuchs and Ferdinand Georg Frobenius investigated linear differential equations of second order and higher with variable coefficients.
Marius Sophus Lie contributed his Lie's Theory of Continuous Groups revealed a connection between techniques which had previously been believed to be disconnected.
Graphical considerations were developed by Karl Hermann Amandus Schwarz, Felix Klein and Édouard Jean-Baptiste Goursat.
Takeo Wada extended these methods to the results of Charles Émile Picard and Jules Henri Poincaré.
Numerical methods were developed by Carl David Tolmé Runge, among others.
Sources
- 1926: E.L. Ince: Ordinary Differential Equations ... (previous) ... (next): Chapter $\text I$: Introductory: $\S 1.1$ Definitions
- 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{VI}$: On the Seashore
- 1952: H.T.H. Piaggio: An Elementary Treatise on Differential Equations and their Applications (revised ed.) ... (previous) ... (next): Chapter $\text I$: Introduction and Definitions. Elimination. Graphical Representation
- 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $1$: How Differential Equations Originate
- 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $3$: The Differential Equation
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 1$: Introduction
- 1977: William E. Boyce and Richard C. DiPrima: Elementary Differential Equations and Boundary Value Problems (3rd ed.) ... (next): Chapter $1$: Introduction
- 1977: A.J.M. Spencer: Engineering Mathematics: Volume $\text { I }$ ... (previous) ... (next): Chapter $1$ Ordinary Differential Equations: $1.1$ Introduction
- 1978: Garrett Birkhoff and Gian-Carlo Rota: Ordinary Differential Equations (3rd ed.) ... (next): Chapter $1$ First-Order Differential Equations: $1$ Introduction
- 1992: William E. Boyce and Richard C. DiPrima: Elementary Differential Equations and Boundary Value Problems (5th ed.) ... (next): Chapter $1$: Introduction: $1.1$ Classification of Differential Equations
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): differential equation
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): differential equation
- 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $8$: The System of the World: The differential equation