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 (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:

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

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