DIFFERENTIAL EQUATIONS AND ITS TYPES

What are the differential Equations?

In our world, things constantly change or evolve and to describe these changes, Differential Equations are used. It is a rather logical and natural way to describe how things change within the universe. A differential equation is a mathematical equation that matches a particular function to a range of its derivatives. They are applied in real life as well in different fields like engineering, economics, and commerce, physics and biology. They are applied in a way where the function displays physical quantities and the derivatives represent their rates of change and together they form a differential equation that defines their relationship with each other. 

How are differential equations used?

They are also used to find a set of functions that satisfy a particular equation. The equation usually contains one function and a list of its derivatives. It can be used to find out different things like changes in population, spring vibration, heat movement, etc. We can use the solutions of these equations to predict future changes by developing trends through graphs. Differential equations can be further classified into different types whereby each type is useful in specific contexts and aspects of a given situation or theory.

What is Ordinary Differential Equations?

An ordinary differential equation contains one unknown function along with its derivatives and a set of known functions (x). The unknown function is shown in the form of a variable denoted by “y” which is dependent upon the known functions (x). X is, therefore, the dependent variable and Y the independent variable. The ordinary differential equations only have one independent variable in the whole equation. These equations are the opposite of Partial Differential Equations. The formula for it is ”y=f(x)”. 

How are ordinary differential equations used?

These equations are widely used in different forms depending upon the use of notations in the tasks they are being used for. 

Partial Differential Equations

A partial differential equation contains more than one unknown function along with their partial derivatives. The unknown functions are usually referred to as multivariable functions. These equations define problems that include more than one factors or variables which can be used to create multidimensional computer models or be solved in a closed-form manner. They contain the rates of change concerning the different variables presented in the equation. The variables usually act as parameters for the given function. 

How is the partial differential equation used?

They can be used in different aspects such as quantum mechanics, heat transfer, elasticity and sound (acoustics). 

Linear Differential Equations

A linear differential equation is an equation in which the solutions may be expressed in an integral form. They are usually ordinary differential equations. However, they can be partial as well if the unknown function is dependent upon more than one variable and the equation contains partial derivatives. A linear differential equation contains a linear polynomial in both the unknown functions as well as its derivatives written in the form of an equation. Holonomic functions are the solutions of linear differential equations with polynomial functions. These contain widely used functions such as exponential functions, sine, and cosine, logarithms, etc. 

How are linear differential equations used?

They can be used to determine limits and numerical evaluation of data under specific error boundaries.

Non-linear Differential Equations

A non-linear differential equation is simply not linear concerning the unknown function and related derivatives. It is rather difficult to get accurate solutions of non-linear equations and to solve them the equation needs to contain specific symmetries. Non-linear equations are also written as linear equations for them to be considered as approximations in order to get accuracy in solutions. These equations are incredibly complex and difficult to solve as they contain many different forms as the functions have more than one parameter for each variable. Therefore, it is usually not possible to know for sure if the numerical solution of that equation is for that particular phenomenon or a result of the discretization process that has been developed for the conversion of continuous functions for numeric evaluation. It is vital to note that the discretization might not be correct and so the solution would be for the model developed and not the actual phenomenon. 

Homogeneous Differential Equations

There can be of two types, either homogeneous first-order a differential equation or homogeneous linear differential equation. A homogeneous equation simply put is when there is a homogeneous function of that unknown function and related derivatives. In the case of first-order differential equations, both functions are homogeneous and of the same degree. 

How to solve homogeneous equations?

These types of equations are solved by separating the equations into two parts and solved through integration. A homogeneous linear equation is one in which each non zero terms of the presented equation should be dependent on either the unknown function or its derivative. These types of equations do not have any constant terms. They can be solved through the integration of the solution by removing the constant term. 

Non-homogeneous Differential Equations

They are different from homogeneous equations in the sense that they can have only constants in the equation. Solutions to the non-homogeneous equations can be written as the sum of the solution to the corresponding homogeneous equation. The non-homogeneous equations can either be an ordinary second-order or partial differential equation. 

How to solve can non-homogeneous differential equations?

The ordinary equation of second-order can be solved through variation of parameters by finding a set of basis vectors for the corresponding homogeneous differential equation. In the case of the non-homogeneous partial differential equation, the solution is generated by applying Lagrange’s method.

Conclusion

After reviewing the whole summary I would like to include that, Differential Equations are applicable in a wide array of disciplines as mentioned above including chemistry, geology and even archaeology. They have over time been used to convert any physical processor phenomenon into a mathematical model to predict future occurrences in that particular aspect. From the control of motion in an airplane to the calculation of the heartbeat of species in science, have played a stellar role in human advancement in most if not all of the disciplines.