Real Life Applications of Differential Equations| Uses Of Differential Equations In Real Life

00:11:11
https://www.youtube.com/watch?v=zm_UqjVLViU

Zusammenfassung

TLDRThe video explains various applications of differential equations in contemporary life, illustrating their significance in numerous fields. It discusses how these equations model population dynamics, such as the logistic growth model and the SEIR model for disease spread, emphasizing their role in pandemic analysis. It also explores their applications in acoustics to understand music, in thermodynamics through Newton's law of cooling, and in radioactive decay to determine half-lives. Additionally, the video covers applications in economic analysis, highlighting how GDP changes over time can be modeled with differential equations, and introduces Maxwell's equations governing electromagnetic fields. The second law of motion is discussed regarding force and mass variations, particularly in scenarios like rocket propulsion. Overall, differential equations emerge as vital tools across disciplines, making complex systems comprehensible and manageable.

Mitbringsel

  • 📊 Differential equations are essential for modeling population growth.
  • 🎶 They help analyze musical acoustics and sound variations.
  • 🌡️ Newton's law of cooling is described using differential equations.
  • ⚛️ Radioactive decay laws can be represented with these equations.
  • 💰 Economic changes such as GDP growth are modeled through differential equations.
  • ⚡ Maxwell's equations govern electromagnetic behavior and technology.
  • 📈 Understanding force-mass relationships in motion relies on differential equations.
  • 🦠 The SEIR model is crucial for tracking infectious disease spread.
  • 🎆 Complex systems in physics are simplified using differential equations.
  • 💡 Differential equations offer insights across various fields and real-world applications.

Zeitleiste

  • 00:00:00 - 00:05:00

    The video discusses the significant real-world applications of differential equations, highlighting their use in population modeling, specifically the logistic population growth model, which describes the changes in populations over time by considering the carrying capacity. Additionally, the SEIR model is presented for analyzing COVID-19, detailing how susceptible, exposed, infected, and removed individuals can be tracked using differential equations, emphasizing their crucial role during the pandemic.

  • 00:05:00 - 00:11:11

    Further applications of differential equations include musical acoustics, Newton's law of cooling, radioactive decay, economic analysis (specifically GDP), Maxwell's equations governing electromagnetism, and Newton's second law of motion, each illustrated with examples such as temperature changes in ice cream, the decay of radioactive elements like cobalt, and the dynamics involved in rocket motion, demonstrating the versatility and importance of differential equations across various fields.

Mind Map

Video-Fragen und Antworten

  • What is the logistic population growth model?

    It is a model where the rate of change of population is proportional to the product of the population and the difference between the carrying capacity and the population itself.

  • How are differential equations used in music?

    They are used to analyze sound waves and their variations, helping to understand why different musical notes sound pleasing together.

  • What does Newton's law of cooling state?

    It states that the loss of heat from a body is directly proportional to the difference in temperature between the body and its surroundings.

  • What is the significance of half-life in radioactive decay?

    Half-life indicates the time required for half of the radioactive atoms in a sample to decay, helping to determine the remaining quantity over time.

  • How do differential equations apply to economics?

    They are used to analyze the change in GDP over time and help address issues like recession and economic growth.

  • What are Maxwell's equations?

    A set of four coupled partial differential equations that describe how electric and magnetic fields are generated and altered.

  • How does the second law of motion relate to differential equations?

    The rate of change of momentum is proportional to the applied force, which can be expressed using differential equations.

  • What is the SEIR model?

    It is a model used to analyze the spread of infectious diseases, incorporating susceptible, exposed, infected, and recovered populations.

  • What do differential equations reveal about pandemic management?

    They provide analytical tools to understand infection rates and recovery, essential for effective pandemic response.

  • Why is understanding sound variation important in music?

    It helps explain why certain harmonies sound pleasant, while others do not.

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Untertitel
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Automatisches Blättern:
  • 00:00:00
    Hello Everyone.today lets explore the real world applications of differential equations
  • 00:00:06
    and friends by the end of this video you will have a good idea
  • 00:00:10
    of the immense contribution of differential equations in our present world. so let's start
  • 00:00:16
    music
  • 00:00:18
    an important application is the population model
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    so friends,whether its the growth of human population
  • 00:00:24
    or the number of predator versus prey or
  • 00:00:26
    the growth of micro organisms
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    we have population which is constantly changing
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    with respect to time and this can be very
  • 00:00:34
    well depicted using differential equations
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    and one such model is the logistic population growth model
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    if p is the population and t is the time then the rate of change of p
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    with respect to t is directly proportional to the product of p and the difference between the carrying capacity and p
  • 00:00:57
    so friends, what is carrying capacity? it is nothing but the maximum sustainable population
  • 00:01:01
    so friends, on solving this differential equation, we get this expression
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    using which, we can find out the population for any given value of time
  • 00:01:11
    here is another model the SEIR Model, that is being currently used to analyse the covid pandemic
  • 00:01:18
    friends, S stands for Susceptible, that is , those people , who are at risk of getting the infection
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    and ds/dt is given by this differential equation
  • 00:01:28
    friends, did you see that beta is the transmission rate of the virus?
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    now, from these susceptible people, we have some people who will be exposed to the virus
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    and that is denoted by E, so de/dt is given by this differential equation and sigma is the infection rate
  • 00:01:47
    now, some of these exposed people will actually get infected with the virus, and that stands for I
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    so here is the differential equation that shows you dI/dt and gamma is the recovery rate
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    and last, but not the least we have R that is removed
  • 00:02:05
    that is those infected people who either died or got completely recovered
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    and this is the differential equation for dr/dt. so friends , this only underlines
  • 00:02:16
    how invaluable this data is for all those people who are currently
  • 00:02:21
    fighting with the pandemic, so differential equations , undoubtedly is very valuable for population models.
  • 00:02:29
    Friends, did you know that differential equations is very useful
  • 00:02:33
    in the world of music? Let's see how
  • 00:02:36
    so here is music from the piano( notes from the piano)
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    and this is from a synthesizer playing the flute sound (notes from a flute)
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    so why does a piano and flute sound different, although they are playing the same set of notes?
  • 00:02:57
    well, this and any more questions are being analysed, thanks to differential equations.
  • 00:03:03
    this is the differential equation that is used for all musical acoustics
  • 00:03:07
    so friends, as you can see, this is a second order and partial differential equation
  • 00:03:12
    so why are we using differential equations over here?
  • 00:03:15
    because we are dealing with sound waves, whose shapes
  • 00:03:18
    are constantly changing with respect to time, and hence we can use
  • 00:03:23
    differential equations to model their behaviour.
  • 00:03:26
    now friends, this differential equation describes sound variations
  • 00:03:30
    and solving this will help us understand why some notes
  • 00:03:34
    when played together sound very pleasing- something like this (music)
  • 00:03:43
    while , others don't ! like this - (sounds)
  • 00:03:54
    the next application is newton's law of cooling.
  • 00:03:57
    so friends, this law states that the loss of heat from a body is directly proportional
  • 00:04:02
    to the difference in the temperature of the body and that of its surroundings.
  • 00:04:07
    so you have temperature of the body that is changing with respect to time,
  • 00:04:12
    so that means we can use differential equations to describe this law.
  • 00:04:16
    on solving this differential equation, we will get a relation between the temperature of the object,
  • 00:04:22
    temperature of the surrounding and time t.
  • 00:04:25
    so friends , you can find the temperature of the object at any given point of time.
  • 00:04:29
    now friends this law is being used widely in the packaging industry
  • 00:04:34
    where food is subject to high temperatures and then it has to be cooled for packaging purposes.
  • 00:04:40
    now lets see how this law works for an ice cream.
  • 00:04:44
    the temperature of an ice cream when taken out of a freezer, supposing is -20 degrees Celsius,
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    and the outside temperature is 30 degrees Celsius.
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    Now after 1 minute its been noted that the temperature of the ice cream has increased to - 10 degrees Celsius.
  • 00:05:01
    You need to find out what is the temperature of the ice cream after 5 minutes.
  • 00:05:06
    surely it has increased, but you can find out, by how much,using newton's law of cooling,
  • 00:05:13
    by doing these calculations, and friends, you will come to know
  • 00:05:16
    that the ice cream temperature after 5 minutes is 15.61 degrees Celsius,
  • 00:05:22
    which means the ice cream is going to literally melt in your hands!!
  • 00:05:27
    differential equations are very useful in radioactive decay. lets see how.
  • 00:05:31
    so friends, radioactive elements like uranium, radium , cobolt, etc on being unstable
  • 00:05:38
    spontaneously decay, to give the daughter nucleus and the alpha, beta or gamma rays,depending on the type of decay.
  • 00:05:47
    so we have a law of radioactive decay, which states that the amount of radioactive element that is decaying per unit time
  • 00:05:56
    is actually directly proportional to the total amount of radioactive element present.
  • 00:06:02
    so since the quantity of the element is changing with respect to time,
  • 00:06:07
    we can use differential equations to express this law.
  • 00:06:11
    and friends please note that the negative sign indicates that
  • 00:06:14
    the quantity of the element is actually decreasing as time progresses.
  • 00:06:19
    on solving this differential equation, we will get the relation between
  • 00:06:23
    the quantity of the element and time t,
  • 00:06:26
    so you can find out how much quantity of the element is remaining at any given point of time.
  • 00:06:33
    one use of this equation is to find the half-life of radioactive elements,
  • 00:06:39
    which is nothing but, the time that is required for half of the atoms in the sample to decay.
  • 00:06:45
    so here we have cobolt, which is a radioactive element, and friends, its half life can be found out to be 5.27 years,
  • 00:06:54
    using appropriate values in the equation.
  • 00:06:58
    This half life value is of great importance friends, because it tells us
  • 00:07:02
    how much quantity of the radioactive element is actually present as time progresses.
  • 00:07:07
    And as you can see in this graph, the percentage of cobolt-60
  • 00:07:12
    that is remaining as time progresses is exponentially decreasing, which means,
  • 00:07:18
    Cobolt, which is widely used for cancer treatment has to be replaced regularly in order to be effective,
  • 00:07:25
    and for this, differential equations is very useful.
  • 00:07:29
    the next application is in economic analysis which includes GDP Calculation.
  • 00:07:33
    Friends the change in GDP with respect to time is found to be directly proportional
  • 00:07:39
    to the current GDP of the economy.
  • 00:07:42
    So if x is the GDP of the economy, here is the differential equation that comes in to the picture.
  • 00:07:48
    g is the growth rate, and on solving this differential equation,
  • 00:07:52
    we will get the relation using which we can find the GDP for any given point of time.
  • 00:07:58
    differential equations come very handy in macro economics,
  • 00:08:02
    which is nothing but the study of large economic systems, like that of a country.
  • 00:08:08
    and one such equation friends, is the fokker-planck equation,
  • 00:08:12
    which is a partial differential equation describing time evolution.
  • 00:08:17
    so friends, equations like this, help us understand why some countries
  • 00:08:21
    are poorer than others, what causes recession and how to deal with it.
  • 00:08:28
    the next application is the maxwell's equations.
  • 00:08:31
    friends, this is a set of 4 coupled, partial differential equations
  • 00:08:35
    that describe how electric and magnetic fields are generated and altered.
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    so the first law, is the Gauss' Law of Electric Field,
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    the second law is the Gauss' magnetism law,
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    the third law is the faraday's law of induction
  • 00:08:51
    while the fourth law is the ampere's law.
  • 00:08:53
    so the triangle symbol that you see is the del operator that is used for divergence and curl vector operations.
  • 00:09:01
    So these are four complex laws, but, they are the building blocks
  • 00:09:05
    for all types of electrical, optical, magnetic technologies and innovations ,
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    such as, smart phones, mobile phones and computer technology,
  • 00:09:15
    making mri scanners in hospitals,
  • 00:09:18
    electric motors and generators,
  • 00:09:21
    and also to better understand the various phenomena of light.
  • 00:09:27
    the last application we will be discussing is the newton's second law of motion.
  • 00:09:31
    so friends, this law states that the rate of change of momentum of an object
  • 00:09:37
    is directly proportional to the force that is applied on it.
  • 00:09:41
    so since momentum is changing with respect to time, that means we can use differential equations.
  • 00:09:47
    friends there are two scenarios to this.
  • 00:09:49
    if we assume the mass of the object to be constant, then we will get the well-known formula
  • 00:09:55
    that force is equal to mass multiplied by acceleration.
  • 00:09:58
    however, if the mass is variable, that is either it is increasing or decreasing with respect to time,
  • 00:10:04
    here is the differential equation that we will be using.
  • 00:10:08
    the variable mass equation is very useful to study the motion of rockets
  • 00:10:12
    because the mass of a rocket is decreasing with time, as fuel is burnt and propellants are being expelled.
  • 00:10:20
    but lets take a simpler example of pushing a shopping cart.
  • 00:10:24
    so friends, if the shopping cart is empty, we have all experienced
  • 00:10:28
    that it is going to take less force to push it, because, the shopping cart has less mass.
  • 00:10:34
    so less mass less force needed.
  • 00:10:36
    however if the same shopping cart was filled with say, gifts,
  • 00:10:41
    although you would be delighted to push them home,
  • 00:10:44
    it is definitely going to take more force to push it because, now the shopping cart has more mass.
  • 00:10:51
    so more mass, more force needed to push the cart.
  • 00:10:55
    so friends, thanks for watching, and I hope you found the video useful.
  • 00:10:59
    If so, do like and share the video,
  • 00:11:02
    please leave your comments in the comment section below, and do consider subscribing to Enjoy Math.
  • 00:11:08
    So, till we meet again, take care, Bye.
Tags
  • differential equations
  • population modeling
  • music acoustics
  • Newton's law of cooling
  • radioactive decay
  • economic analysis
  • Maxwell's equations
  • pandemic management
  • SEIR model
  • physics