Differential Equations Definition, Types

In mathematics, a differential equation is an equation that comprises one or more functions with their derivatives as part of the solution. The derivatives of a function are used to determine the rate of change of a function at a given point in space. It is mostly employed in subjects such as physics, engineering, biology, and other related fields. This is accomplished mostly through the investigation of solutions that fulfil both the equations as well as the attributes of those solutions. 

Differential equations: 

It is possible to have one or more terms and the derivatives of one variable (the dependant variable) with regard to the other variable in a differential equation, which is called a differential equation with two variables (i.e., independent variable) 

dy/dx = f(x) 

This example uses “x” as an independent variable while “y” serves as the dependent variable.

As an illustration, dy/dx = 5x.

Depending on the type of differential equation being considered, there may be either partial or ordinary differential equations. With the derivative, you can represent the rate of change, and with the differential equation, you can explain a relationship between a variable that is continuously varying and the rate of change in another quantity. There are a plethora of differential equation formulas that can be used to obtain the derivatives solution. 

Order of differential equations: 

It is the order of the differential equation that corresponds to that of the highest order derivative that is contained in the equation. Here are some examples of differential equations with different orders of operation that are provided. 

• dy/dx = 3x + 2 , The order of the equation is 1
• (d2y/dx2)+ 2 (dy/dx)+y = 0. The order is 2
• (dy/dt)+y = kt. The order is 1. 

First order differential equation: 

For example, as you can see in the first example, it is a first-order differential equation with a degree of one. Unless otherwise stated, all linear equations in the form of derivatives are in the first order of equations. X and y are the two variables in this equation, and only the first derivative (dy/dx) is present. This equation is represented as follows: 

dy/dx = f(x, y) = y’ 

Second Order differential equation: 

The differential equation with a second-order derivative is referred to as the second-order differential equation. It is represented as follows: 

d/dx(dy/dx) = d2y/dx2 = f”(x) = y” 

Degree of differential equation: 

It is known as the degree of a differential equation when it has a power greater than one in its highest order derivative. The original equation is represented as a polynomial equation with derivatives such as y,y”, and y”’ and so on.

Assume that (d²y/dx²)+ 2 (dy/dx)+y = 0 is a differential equation, in which case the degree of this equation is 1. Here are a few more illustrations: 

• Dy/dx + 1 = 0, degree is 1
• (y”’)3 + 3y” + 6y’ – 12 = 0, degree is 3
• (dy/dx) + cos(dy/dx) = 0; it is not a polynomial equation in y′ and the degree of such a differential equation can not be defined. 

Types of differential equations: 

Differential equations can be classified into various categories, the most common of which are 

• Ordinary Differential Equations
• Partial Differential Equations
• Linear Differential Equations
• Nonlinear differential equations
• Homogeneous Differential Equations
• Nonhomogeneous Differential Equations 

Ordinary differential equations: 

Known as ODEs in mathematics, the phrase “Ordinary Differential Equations” (also known as ODE) refers to a mathematical equation that has only one independent variable and one or more derivatives with regard to the independent variable. Or to put it another way, the ODE is represented as the relationship between one independent variable x and one real dependent variable y, together with some of their derivatives. 

y’,y”, ….yn ,…with respect to x. 

Partial differential equation: 

It is considered to be linear if the partial differential equation (PDE) f is a linear function of the variable u and its derivatives. The simple PDE is provided by the expression; 

∂u/∂x (x,y) = 0  

It follows from the preceding relationship that the function u(x,y) is independent of x, which is the simplified form of the partial differential equation formula previously stated. The order of a partial differential equation (PDE) is the order of the highest derivative term in the equation. 

Linear differential equation: 

One way to think about a linear differential equation is as a linear polynomial equation, which is made up of the derivatives of several separate variables. When the function is dependent on variables and the derivatives are partial, the equation is referred to as Linear Partial Differential Equation (LPDE).

P and Q are either constants or functions of the independent variable (in this case x), respectively, in a differential equation of the above form. This type of differential equation is known as the first-order linear differential equation. 

Homogeneous differential equation: 

It is considered to be a homogeneous differential equation when the degree of f(x,y) and g(x,y) are the same in the form of f(x,y)dx = g(x,y)dx. A function of the type F(x,y) that can be expressed in the form kn F(x,y) is referred to as a homogeneous function of degree n, where k0 is the number of variables in the function. As a result, the homogeneous functions of the same degree of x and y are denoted by f and g. In this case, the change in the variable y = ux results in an equation of the type 

dx/x = h(u) du

Which could be easily integrated. 

Conclusion: 

Differential equations have a wide range of applications in a variety of domains, including applied mathematics, physics, and engineering, among others. Aside from technical applications, they are also employed in the solution of a wide range of real-world problems. In addition to the numerous engineering applications, it may be used to comprehend the motion of waves in physics and heat conduction analysis, among other things. The ordinary differential equation can be used in the engineering industry to determine the relationship between various elements of a bridge, which is a common application.