Definition:Differential Equation

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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 is a differential equation which has exactly one independent variable.

All the derivatives occurring in it are therefore ordinary.


The general ordinary differential equation 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 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 Nonlinear

Differential equations can also be classified as to whether they are linear or nonlinear.


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.


Hence a linear differential equation is of the form:

$\map {P_0} x y + \map {P_1} x \dfrac {\d y} {\d x} + \map {P_2} x \dfrac {\d^2 y} {\d x^2} + \cdots + \map {P_n} x \dfrac {\d^n y} {\d x^n} = \map Q x$

where $P_0, P_1, \ldots, P_n, Q$ are functions of $x$.


Nonlinear

A nonlinear 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:

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.


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

First Order Linear Ordinary Differential Equation

Ordinary linear differential equation of the $1$st order:

$\dfrac {\d y} {\d x} = x$


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

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