Differential Equations Applications

Application Of Differential Equation In Mathematics

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.

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Significance of Differential Equations

A significant magnitude of differential equation as a methodology for identifying a function is that if we know the function and perhaps a couple of its derivatives at a specific point, then this data, along with the differential equation, can be utilized to effectively find out the function over the whole of its domain.


Types of Differential Equations

A Differential Equation exists in various types with each having varied operations. 

  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

As a consequence of diversified creation of life around us, 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 differential equation. So, let’s find out what is order in differential equations.


Order Of Differential Equation

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 basically 2 types of order:-

  1. First order differential equation

  2. Second order differential equation


Application Of First Order Differential Equation

Modeling is an appropriate procedure of writing a differential equation in order 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.:

  1. Population Problems

  2. Falling Objects

  3. 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 solution.


Application Of 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 a number of processes in physics. Applications of differential equations in engineering also have their own 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 

Problem

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

y′′′+y2+ey′=0. 

Solution 

Given that, 

y′′′+y2+ey′=0 

Thus, (dx3d3y​)+y2+edxdy​=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 equation we have will be –3​ 


Fun Facts

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

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

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

FAQ (Frequently Asked Questions)

1. Can Differential Equations Be Applied In Real Life?

YES! And the amazing thing is that differential equations are applied in most disciplines ranging from medical, chemical engineering to economics. That said, you must be wondering about application of differential equations in real life. So, since the differential equations have an exceptional capability of foreseeing the world around us, they are applied to describe an array of disciplines compiled below;-

  • explaining the exponential growth and decomposition

  • growth of population across different species over time

  • modification in return on investment over time

  • find money flow/circulation or optimum investment strategies

  • modeling the cancer growth or the spread of a disease

  • demonstrating the motion of electricity, motion of waves, motion of a spring or pendulums systems

  • modeling chemical reactions and to process radioactive half life

2. Why Are Differential Equations Useful In Real Life Applications?

Nearly any circumstance where there is a mysterious volume can be described by a linear equation, like identifying the income over time, figuring out the ROI, anticipating the profit ratio or computing the mileage rates. Many people make use of linear equations in their daily life, even if they do the calculations in their brain without making a line graph.

One of the fundamental examples of differential equations in daily life application is the Malthusian Law of population growth. dp/dt = rp represents the way the population (p) changes with respect to time. The constant r will alter based on the species. Malthus executed this principle to foretell how a species would grow over time.