# 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.

## 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

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 $\Phi$.

## 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 it can be written in the form:

- $y^{\left({n}\right)} = f \left({x, y, y', y'', \dots, y^{\left({n-1}\right)}}\right)$

### 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:

- $\dfrac {\d y} {\d x} = 1 + x y^2$

### 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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