Differential Equations Applications

Within mathematics, a differential equation refers to an equation that brings in association one or more functions and their derivatives. In applications, the functions usually denote the physical quantities whereas the derivatives denote their rates of alteration, and the differential equation represents a relationship between the two.

A relationship between two quantities, two functions, two variables, or a collection of variables, or two functions is represented by an equation. 

A differential equation is a series of formulas that describes the connection between a function and its derivatives. Numerous instances demonstrate the application of these equations.

The functions are the ones that signify some sort of operation, the rate of change during that operation is the derivative of that operation, and the differential equation is the relationship between them. 

These equations are written in terms of degree order, such as first-order, second-order, and so on. Its applications are prevalent in engineering, physics, and other fields.

Types of Differential Equations Application

Below are the types of differential equations:

1. Ordinary Differential Equations.

2. Partial Differential Equations

3. Linear Differential Equations.

4. Non-linear differential equations.

5. Homogeneous Differential Equations.

6. Non-Homogenous Differential Equations

Ordinary Differential Equations

The "Ordinary Differential Equation," or ODE, is a mathematical equation with only one independent variable and one or more derivatives concerning that variable. 

The ordinary differential equation is thus represented as a relation with one independent variable x and one real dependent variable y, as well as some of its derivatives y', y",....yⁿ,... concerning x.

There are two types of ordinary differential equations: homogeneous and non-homogeneous.

Homogeneous Differential Equation

A homogeneous differential equation is a differential equation in which all of the terms have the same degree.

In general, they can be represented as:

P(x,y)dx + Q(x,y)dy = 0,

The homogeneous functions P(x,y) and Q(x,y) are both of the same degrees.

Nonhomogeneous Differential Equation

A non-homogeneous differential equation is a differential equation in which the degree of all terms is not the same.

Partial Differential Equations

A partial differential equation, or PDE, is an equation involving only partial derivatives of one or more functions of two or more independent variables. Eg,

\[\frac{\partial u}{dx}\] + \[\frac{\partial}{dy}\] = 0

\[\frac{\partial^{2} u}{\partial x^{2}}\] + \[\frac{\partial^{2}u}{\partial x^{2}}\] = 0

Linear Differential Equation

An equation with a variable, its derivative, plus a few other functions is known as a linear differential equation.

A linear differential equation's typical form is dy/dx + Py = Q, which includes the variable y and its derivatives.

In this differential equation, P and Q are either numeric constants or x functions.

As a consequence of the diversified creation of life around us, a multitude of operations, innumerable activities, therefore, differential equations, to model the countless physical situations are attainable. 

The classification of differential equations in different ways is simply based on the order and degree of the differential equations. So, let’s find out what is the order in differential equations.

Order of Differential Equation

The order of a differential equation represents the order of the highest derivative which subsists in the equation. Actuarial Experts also name it as the differential coefficient that exists in the equation. There are 2 types of order:-

• First-order differential equation

• Second-order differential equation

Applications of Differential Equations

Ordinary differential equations are used in the real world to calculate the movement of electricity, the movement of an item like a pendulum, and to illustrate thermodynamics concepts. Graphic representations of disease development are another common usage for them in medical terminology.

Differential equations can be used to describe mathematical models such as population expansion or radioactive decay.

• First Order Differential Equation

Modeling is an appropriate procedure for writing a differential equation to explain a physical process.

Almost all of the differential equations whether in medical or engineering or chemical process modeling that are there are for a reason that somebody modeled a situation to devise with the differential equation that you are using.

Now let’s know about the problems that can be solved using the process of modeling. For that we need to learn about:-

Modeling with First Order Differential Equation

Here, we have stated 3 different situations i.e.:

• Population Problems

• Falling Objects

• Mixing Problems

In each of the above situations, we will be compelled to form presumptions that do not precisely portray reality in most cases, but in absence of them, the problems would be beyond the scope of the solution.

• Second-Order Differential Equation

A second-order differential equation involves the unknown function y, its derivatives y' and y'', and the variable x. Second-order linear differential equations are employed to model many processes in physics. 

Applications of differential equations in engineering also have their importance.

Models such as these are executed to estimate other more complex situations.

Modeling with Second Order Differential Equation

Here, we have stated 3 different situations i.e.:

• Harmonic Motion

• Bonds between atoms or molecules

• Systems of the electric circuit consisted of an inductor, and a resistor attached in series.

Solved Example 

Problem1. Find out the degree and order of the below-given differential equation.

y′′′+y²+ey′=0. 

Ans: Given that, 

y′′′+y²+ey′=0 

Thus, (dx³d³y​)+y²+e.dxdy​=0 

The degree of a differentiated equation is the power of the derivative of its height. However, the above cannot be described in the polynomial form, thus the degree of the differential equation we have is unspecified.

Considering the number of height derivatives in a differential equation, the order of differential equations we have will be –3​. 

Fun Facts

1. How do Differential equations come into existence? With the invention of calculus by Leibniz and Newton.

2. Applications of differential equations in physics also have their usage in Newton's Law of Cooling and Second Law of Motion.

3. Only if you are a scientist, chemist, physicist or biologist—can have a chance of using differential equations in daily life.