What is differential equation?

#1

A Differential Equation (DE) is an equation containing the derivatives of one or more dependent variable(s) with respect to one or more independent variable(s) or

A differential equation is an equation relating some function f to one or more of it's derivatives. (Krantz - Differential Equations Demystified).

Examples of DEs:

1. $x\frac{dy}{dx} + y = x$ ,

2. $\frac{d^{2}y}{dt^{2}} + 5\frac{dy}{dt} + 10y = 0$ ,

3. $\frac{d^{2}y}{dx^{2}} + xy\left(\frac{dy}{dx} \right)^{3} = 0$ ,

4. $\frac{d^{5}y}{dx^{5}} + 10\frac{d^{3}y}{dx^{3}} + 5x = \text{sin}y$ ,

5. $\frac{\partial v}{\partial s} + \frac{\partial v}{\partial t} = v$ ,

6. $\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} + \frac{\partial^{2} u}{\partial z^{2}} = 0$ .

Classification by type

Under this classification we have Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs).

A DE containing only ordinary derivatives of one or more dependent variable(s) with respect to a single independent variable, is said to be an ODE.

Examples of ODEs are equations (1 - 4).

A DE involving partial derivatives of one or more dependent variable(s) with respect to more than one independent variables is called a PDE.

Examples of PDEs are equations ( 5 & 6).

Differential Equations can further be classified by order, degree and linearity.

Classification by order, degree and linearity

Both ODEs and PDEs can be classified depending on the order of the highest derivative appearing in the equation.

The order of a DE (ODE or PDE) is the order of the highest derivative in the equation.

Degree of a DE is the power of the highest derivative in the equation.

Examples: equation (1) is a first-order ODE, equations (2) and (3) are second-order ODEs, equation (5) is a first-order PDE, and equation (6) is a second-order PDE.

Next, we consider the classification by linearity of the general $n^{th}$ -order ODE which is often represented by

$F(x,y, y', \dots, y^{n-1}, y^{n}) = 0.$

An ODE of order $n$ in the dependent variable, say $y$ , and the independent variable, say $x$ , is linear if it can be written in the form
$a_n(x)\frac{d^{n}y}{dx^{n}} + a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}} + \dots + a_1(x)\frac{dy}{dx} + a_0(x)y = b(x),$

where $n$ is not identically zero.

Generally, for linear ODEs:

(a) The dependent variable $y$ and all its derivatives are of the first degree only; that's, the power of each term in an equation is 1.

(b) Each coefficient depends only on the independent variable $x$ .

(c) Functions of $y$ such as sin( $y$ ), cos( $y$ ), $e^{y}$ or functions of derivatives of $y$ such as $e^{y'}$ cannot appear in a linear ODE.

Examples of linear ODEs are equations (1) & (2).

Examples of non-linear ODEs include equation (3).

Linear ODEs can also be classified according to the nature of coefficients of the dependent variables and their derivatives. For examples, ODE (2) is linear with constant coefficients while ODE (1) is linear with variable coefficient.

An ODE that is not linear is said to be linear.

Differential Equations (DEs)As Mathematical Models

When Mathematical modelling is used to describe physical, chemical or biological phenomena in sciences and engineering, one of the most common results of the process is a system of ODEs or PDEs.

DE problems involve objects that obey certain scientific laws. The laws involve rates of change of one or more quantities with respect to other quantities. These rates of change are mathematically expressed in terms of derivatives.

In the next lessons, we will explore several examples of DEs models that originate from the real-world problems.

Solutions to ODES

Given an $n^{th}$ -order ODE $F(x, y, y', \dots, y^{n}) = 0$ ,

its solution is a function $\Phi$ that possesses at least $n$ derivatives and satisfies

$F(x, \Phi(x), \Phi'(x), \dots, \Phi^{n}(x)) = 0$ for all $x$ in interval I.

Solution to DE can be implicit or explicit.

A solution to DE in which the dependent variable is expressed solely in terms of the independent variable(s) and constant is said to be an explicit solution. We can think of an explicit solution as an explicit formula, $y = \Phi(x)$ .

An implicit solution to DE is a relation $G(x,y) = 0$ on an interval $I$ provided there exists at least one function $\Phi$ that satisfies the relation as well as the given equation on the interval $I$ .

#2

Differential equation is the function obtained as the result of differentiation
$f'(x)=x^{2}+12x-3=0$ is the differential equation

#3

John Linus B, wrote:Differential equation is the function obtained as the result of differentiation
$f'(x)=x^{2}+12x-3=0$ is the differential equation

I think, you must emphasize that a differential equation is a relationship between some function(s) and its derivatives.

#4

Daaaah

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What is differential equation?

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What is differential equation?

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