what is a first order differential equation?

Mathematical models for real world phenomena often take the form if equation relating various quantities and their rates of change. For example the motion of particle involves the distance covered in time t and velocity v or acceleration a. Now the rate of change `\frac{ds}{dt}`of s with respect to t is the velocity v and rate of change `\frac{dv}{dt}` of velocity with respect to t is the acceleration a. A particle moving in a straight line has an equation of motion s=f(t) where t is in second and s is in meters Velocity satisfy the equation.

`V= `\frac{ds}{dt}` =4t²+5t-3`

 This lead us to the definition of differential equation.

Definition of first order differential equation

Any equation involving one dependent variable and its derivative with respect to one or more independent variables is called differential equation

For example

• `\frac{dy}{dx}`+ycosx    = sinx
• `\frac{d^2y}{dx^2}` + xy`\left(\frac{dy}{dx}\right)^2`  = 0

Types of differential equation

There are some type of differential equation.

• ·         Partial differential equation
• ·         Ordinary differential equation
• ·         Linear differential equation
• ·    Non linear differential equation

Partial differential equation

A differential equation involving partial derivative of the dependent variable with respect to more than one independent variable is called partial differential equation.

Example of Partial differential equation

1. x`\frac{\partial z}{\partial x}` + y`\frac{\partial z}{\partial y}` = nx
2. `\frac{\partial^2u}{\partial x^2}` + `\frac{\partial^2u}{\partial y^2}` + `\frac{\partial^2u}{\partial z^2}`

Define Ordinary differential equation

A differential equation in which derivative of the dependent variable with respect to a single  independent variable occur is called a ordinary differential equation.

Example Ordinary differential equation

`\frac{d^2y}{dx^2}` + xy`\left(\frac{dy}{dx}\right)^2`  = 0

Linear differential equation

 

An ordinary differential equation

F`\left(x,y,\frac{dy}{dx},\frac{d^2y}{dx^2},...,\frac{d^ny}{dx^n}\right)` = 0

is said to be linear if F is a linear function of variables

`x,\frac{dy}{dx},\frac{d^2y}{dx^2},...\frac{d^ny}{dx^n}`

a related definition applies to partial differential equations.

So the linear  ordinary differential equation of order n is

`a_0\left(x\right)\frac{d^ny}{dx^n}\;+\;a_1\left(x\right)\frac{d^{n-1}y}{dx^{n-1}}\;+...+\;a_{n-1}\left(x\right)\frac{dy}{dx}\;+\;a_n\left(x\right)y\;=F\left(x\right)`

where `a_0\left(x\right)` is not identically zero.

It should be carefully note able that in a linear ordinary differential equation

•      The dependent variable y and its derivative have degree one.

·         there is no product of y or any of it derivatives appears.
·         There is no transcendental functional y or its derivative  occur.

Non linear differential equation

A differential equation is said to be nonlinear differential equation when it is not linear.

Application of differential equation

differential equation occur in the mathematical formulation of many problems in science and engineering. some such problems are

• To decide the motion of projectile, rocket, satellite or planet.
• To locate the charge or current in an electric circuit.
• Study of chemical reactions.
• Determination of curves with given geometrical properties.

Degree and order of differential equation

 Order of differential equation

The order of a differential equation is the order of the higher derivative that occur in the equation

Degree of differential equation

The degree of differential equations the greatest exponent of the highest order derivative that appear in the equation.