Applications of Differential Equations in Real Life

 Introduction

Differential equations are mathematical models that describe how variables change over time. They play a crucial role in predicting behaviors across various scientific fields. In this post, we’ll explore real-life applications in population growth, physics, epidemiology, and economics.

1. Population Growth

Differential equations are used to model population growth, factoring in birth and death rates. For example, the logistic growth equation helps predict population trends by accounting for carrying capacity.

Image: Population growth

2. Physics

In physics, differential equations explain natural laws. Newton’s second law is a perfect example, where the motion of objects can be predicted using second-order differential equations. They also describe heat transfer, wave propagation, and electrical circuits.

3. Epidemiology

Differential equations help track the spread of diseases. The SIR model divides the population into susceptible, infected, and recovered groups to simulate how diseases like COVID-19 spread. This model aids in health planning and pandemic control.

Image: SIR model (source)

4. Economics

In economics, differential equations model growth rates in markets and economies. For instance, the Solow growth model helps economists understand the long-term effects of capital, labor, and technology on economic output.

Image: Solow Growth model (source)

Conclusion
Differential equations are invaluable for predicting dynamic systems in real life. Whether it’s understanding population trends, modeling physical phenomena, tracking diseases, or analyzing economies, these equations help in problem-solving and decision-making.

"Minus times minus is plus"

“How (-)×(-) = + ”?

As we all know that (plus × plus) is plus and it’s easy to understand but have you ever wondered that how (minus × minus) is plus.

Many of us are using this condition in our daily mathematical life knowingly and unknowingly but you may have never thought that how this product of two negative become a positive. Well, I’m going to show you logically how this “-x-‘’ become “+” in a easy way .There might be many other proofs regarding this awesome thing but I’m going  to explain in a simple ways.

Firstly, you must have the knowledge of simple things like this (+ x-= -) and (-x+ =-)

First, let us start with easiest way, (besides mathematics is all about the learning of patterns).

(-1) x (+3) =-3

(-1) x (+2) =-2

(-1) x (+1) =-1

(-1) x (0) = 0

(-1) x (-1) =+1

(-1) x (-2) =+2

(-1) x (-3) =+3

               From the above pattern it’s easy to understand how negative times negative is positive. It sounds total mathematical? How do you feel?

Here we can go with proof which is kind of language type.

Let’s suppose a town, consider the good guys as positive (+) or bad guys as negative (-) in the town. If the guys enter the town take that as positive (+) and if the guys leave the town take that as negative (-):

Now again we go through the following pattern by considering the above assumption:

•        If the good guys (+) enter (+) the town, that is good (+) for the town.            (+) x (+) =+

•        If the good guys (+) leave (-) the town, that is bad (-) for the town.                   (+) x (-) =-

•        If the bad guys (-) enter (+) the town, that is bad (-) for the town.                      (-) x (+) =-

•        If the bad guys (-) leave (-) the town, that is good (+) for the town.                     (-) x (-) =+

So, these were the basic two proofs!

Here are some other which will help you in different way to understand the problem!

We all know that

                         -1+1=0.

 Multiplying this equation by -1, we get:

(-1)*[(-1)+1]=(-1)*(0).

Using the distributive property, we get,                      

(-1)(-1)+(-1)1=(-1)0                                                  [ a(b+c)=ab+ac (distributive property)]

As we know,

                         (-1)1=-1 and (-1)0=0

So, by applying this and adding 1 on both sides,

                         (-1)(-1)+(-1)+1=0+1

                   Or, (-1)(-1)+0=1

                   Or, (-1)(-1)=1

This proves the relationship.

At last but not the least:

From the index rule.

 We know that

(a^x)^y = a^{x y},

Which holds true for a=2 and x=y= -1.

Therefore,

(2^-1)^{-1 =} (2)^(-1)(-1).

By the definition of negative exponent, we get

(2^-1)^-1 =(1/2)^-1  =2.

We can then conclude that 2¹=2^(-1)(-1)

•       1=(-1)(-1)

As we know if the bases are equal, so must the exponents be equal.

These were the illustration that proves how the product of two negative becomes positive.

If you have got any other ways you can comment down.

Thank you!

Twin Prime Conjecture

Primes and Twin Prime Conjecture

Mathematics is one of the most interesting part of science. Usually Math deals with study of patterns that is, number theory is the study of patterns of numbers, geometry is study of pattern of shapes, algebra is study of pattern of putting things together, trigonometry is measurement of shapes, calculus is patterns of continuous motion and change, probability theory is pattern of repetition in random events, statistical theory is pattern of real-world data, and logic is pattern of abstract reasoning.

Primes

What are Primes? Are they really important part of mathematics?

 Yes, primes are really important part of mathematics and are also known as building block of mathematics. A question may arise why primes are building block instead of natural number or real number? The answer is that every integer greater than 1 is either prime or can be expressed as product of prime number. That is every number you see around is either prime or is product of prime that’s why primes are building block of mathematics as elements are building blocks of chemistry.

For example, let us take random numbers: (To show every number is unique decomposition of primes)

1200=2*2*2*2*3*5*5

28=2*2*7

Euclid

The concept of primes is that “A number greater than one is said to be prime if it is divisible by one and number itself.”  A few examples of prime numbers are 2,3,5,7,11 and so on. The history of prime shows earliest surviving records of explicit study of prime number comes from Ancient Greek Mathematics.

Among the small number’s primes are very common. Of the numbers 2 to 20, the numbers are 2,3,5,7,11,13,17,19 are prime a total of eight out of nineteen. The remaining numbers are all composite; that is, they are not prime, since each number can be evenly divided by some smaller numbers (apart from 1). As we look at larger and larger numbers the primes appear to be thin out. While there are 5 primes below 10, there are only 24 below 100 and just 168 below 1000. If we see the average rate at which prime appears, they have an average rate of 0.5 below 10, 0.24 below 100 and just 0.168 below 1000. The table of prime and average up to 1 million is:

N	10	100	1000	10000	100000	100000
Average	0.5	0.24	0.168	0.123	0.096	0.078

The farther we go the smaller the average becomes. A few questions that arises over here are;

·       Does this thinning continue?

·       Or do we reach a point where it reverses and we find a lot of primes?

·       Or do we reach a point where we do not find primes at all?

These questions were answered by Euclid around 300 B.C. He proved that the primes continue forever and there are infinitely many of them, which is also known as Euclid’s Theorem.

Few facts about Primes:

1.       Prime numbers have only two factors 1 and itself.

2.       The only even number that is prime is 2.

3.       There are infinitely many prime numbers.

4.       Every number can be uniquely expressed as product of prime numbers.

5.       The prime numbers cannot be formed by product of two natural number that are both smaller than it.

6.       No even number greater than 2 can be prime as it can be written as the product of 2*n/2.

7.       The largest prime found till date has 23,249,425 digits.

Twin prime Conjecture:

Conjecture in mathematics is a specific statement that is thought to be true but has not been proven yet. Its similar to hypothesis. The main difference between hypothesis and conjecture is that every hypothesis can be tested but not every conjecture can be tested. A conjecture can become first a hypothesis then a theory then at last a law. So, there are many Conjectures in mathematics which are unsolved and there are conjectures also which has been solved (Poincare Conjecture is one of the famous Solved conjecture). Twin prime is also one of the unsolved Conjecture in Mathematics.

Twin prime is a prime number that is either two less or greater than another prime. In other words, twin prime is a prime that has a prime gap of two. for example, member of the twin primes are 17 and 19 The first few twin primes are:

(3,5), (5,7), (11,13), (1719,), (41,43), (59,61), (71,73), (101,103), (107,109), (137,139), … Usually (2,3) is not considered as twin primes.

The Questions of Twin Prime Conjecture are:

1.)    Are there infinitely many twin primes?

2.)    If there are finitely many, which is the largest twin prime?

The work of Yitang Zhang in 2013, as well as work by James Maynard and others has made substantial progress towards proving there are infinitely many twin primes but still it remains to Solved!

This is an open Question that has remained unsolved, So its up to you also!

yitang zhang

james maynard

Best 15+ math quotes

So here are the most intresting and beautiful Math quotes!

Inspiring Math Quotes:

1.     “Do not worry about your difficulties in mathematics. I can assure you mine are still greater” -Albert Einstien

2.     Mathematical works consist of Proofs, just as poems do consist of Character! -Vladimir Arnold

3.     Without Mathematics you can do nothing. Everything around you is mathematics and every thing around you is Numbers.

4.     The only way to learn Mathematics is to do mathematics. – Paul Halmos

5.     Obvious is the most dangerous word in Mathematics.

6.     Mathematics is most misunderstood subject Even by great mathematician. -Keith Davelin

7.     One should simply read mathematics because it helps to arrange one’s idea. -Unknown

the most beautiful one!

8.     The difference between Mathematician and Politician is that mathematician tries to say the most in least words whereas politician do the exact opposite. -Unknown

9.     Arithmetic is being able to count up to twenty without taking off your shoes. -Mickey Mouse

10.  I advise my students to listen carefully the moment they decide to take no more math courses. They might be able to heat the sound of closing doors. – James Caballero

11.  Just because we can’t find a solution it doesn’t mean that does not exist. -Andrew Wiles

12.  Mathematics is key gate to the science. -Roger Bacon

13.  Mathematics is not about numbers, equations, computations or algorithms its about understanding. -William Paul Thurston

14. There should be no such thing as Boring in mathematics. -Edsger Dijkstra

15.  If you stop general math, then you will make general money.

16.  Last but not least “Mathematics is Key and Door to the Sciences.” – Galileo Galilei.

This was top 15 math quotes I liked if you have any more you can add on comment section. For more math Quotes follow us by email and don't miss our uploads.

Domain Codomain and Range

In this article in short, we will talk about domain, codomain and range of a function. In previous article we have talked about function and its type, you can read this here.

Domain, Codomain and Range:

Domain:

The set of input values for which the function is defined is called Domain of a function. That is the function provides output for every elements of domain. For simplicity we can say a set of grains that is

G= {Barley, oats, rice, wheat, …} is a set of domain. Since when you take set G to certain function say mill you will get corresponding flour as output that is wheat flour etc.

Mathematically, if a function is defined from set X to Y then the set X is domain.

So, in the function f(x)=2x from set A to set B;

Fig: Function from A to B

The domain is X={1,2,3,4,5}

Codomain and Range:

In similar manner the Codomain is set of outputs of a function. Mathematically if we define a function from set A to B then the co-domain is set of elements of B for which there is a preimage in A. In the following figure

Fig: Function from A to B

Codomain={2,4,6,8,10}

Where as Range is set of elements in B. That is

Range={2,4,6,8,10,12}.

So, this was an article about Domain, Codomain and Range. Hope you all like this. Don’t forget to follow our blog by email to get email about what we update.