What are the differential equations? Types of Differential Equations

Historical background

Differential equations have been a common topic since their introduction with the invention of the infinitesimal calculus by Newton and Leibniz in the mid-17th century. This had already been proven in much of pure and applied mathematics at the end of the 19th century. Differential equations have played a key role in many disciplines, including physics, biology, engineering and economics.

Application of differential equations

  • Exponential growth and decline
  • Population growth
  • Movement of objects falling by gravity with air resistance and movement of objects suspended by a spring
  • Newton’s Law
    on Refrigeration
  • Article
    goes with the curve
  • Electric circuits
  • Information Technology

What are differential equations?

An equation that contains at least one derivative of a function is called a differential equation. Below are some examples of differential equations.

Before proceeding, it is important to know the basic concepts, such as the order and degree of a differential equation, which can be defined as follows,

i. The order is the highest derivative of a differential equation, e.g. B,

The above differential equation
has only the first derivative, which is , so it is called first-order differential equation
.

Check another differential equation,

In this example, the differential equation has the second derivative, so , so it is called a second order differential equation.

ii. For example, a diploma is B. the exponent of the highest derivative of a differential equation,

In this example, the highest derivative is one and the exponent is also one, so it is a differential equation of first order and first degree. Exactly the same,

Here the highest derivative is 2 and the exponent is 3, hence it is called an ordinary differential equation of degree 2 and degree 3.

Types of differential equations

  1. The ordinary differential equation
  2. partial differential equation
  3. Linear differential equation
  4. Non-linear differential equation
  5. Homogeneous differential equation
  6. inhomogeneous differential equation

For a detailed description of each type of differential equation, see below: – –

1 – Ordinary differential equation

It is a differential equation that contains one or more ordinary derivatives, but no partial derivatives. The ordinary differential equation differs from the partial differential equation in that some independent variables have partial derivatives attached to them, whereas in the differential equation there is only one independent variable, such as y. Newton’s second law of motion is a simple example of an ordinary differential equation.

Another example of the ordinary differential equation
;

2 – Partial differential equation

A partial differential equation is a differential equation that contains partial derivatives. It consists of two or more independent variables. For example, for example..,

3 – Linear differential equation

In terms of the dependent variable(s) and their derivatives, this can be expressed essentially as follows

, where p and q may be constants or functions of the independent variable x.

4 – non-linear differential equation

It is of the second degree or more in terms of dependent variables and their derivatives. For example, for example..,

5 – Homogeneous differential equation

This is a first-order differential equation that can be written as follows

Here f and g are homogeneous functions of the same degree x and y.

For example,
,

6 – Inhomogeneous differential equation

This is a differential equation with a non-zero right-hand side. In this form the inhomogeneous equation 2. can be written in the following order

For example, for example..,

See also : Types of equations

Types of Differential Equations

frequently asked questions

What are the types of differential equations?

Differential equations (definition, types, order, degree …

What is a linear and nonlinear differential equation?

Linear means that the variable in the equation occurs only to the power of one. So, X is linear, but X2 is nonlinear. … If in a differential equation the variables and their derivatives are multiplied only by constants, then the equation is linear.

What is the order of the differential equation?

The order of a differential equation is determined by the derivative of a higher order; the degree is determined by the highest power on the variable. The higher the order of the differential equation, the more random constants must be added to the total solution.

 

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