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Hello Everyone.today lets explore the real world applications of differential equations
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and friends by the end of this video you will have a good idea
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of the immense contribution of differential equations in our present world. so let's start
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music
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an important application is the population model
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so friends,whether its the growth of human population
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or the number of predator versus prey or
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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
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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
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so friends, what is carrying capacity? it is nothing but the maximum sustainable population
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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
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here is another model the SEIR Model, that is being currently used to analyse the covid pandemic
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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
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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
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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
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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
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how invaluable this data is for all those people who are currently
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fighting with the pandemic, so differential equations , undoubtedly is very valuable for population models.
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Friends, did you know that differential equations is very useful
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in the world of music? Let's see how
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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?
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well, this and any more questions are being analysed, thanks to differential equations.
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this is the differential equation that is used for all musical acoustics
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so friends, as you can see, this is a second order and partial differential equation
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so why are we using differential equations over here?
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because we are dealing with sound waves, whose shapes
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are constantly changing with respect to time, and hence we can use
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differential equations to model their behaviour.
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now friends, this differential equation describes sound variations
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and solving this will help us understand why some notes
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when played together sound very pleasing- something like this (music)
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while , others don't ! like this - (sounds)
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the next application is newton's law of cooling.
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so friends, this law states that the loss of heat from a body is directly proportional
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to the difference in the temperature of the body and that of its surroundings.
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so you have temperature of the body that is changing with respect to time,
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so that means we can use differential equations to describe this law.
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on solving this differential equation, we will get a relation between the temperature of the object,
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temperature of the surrounding and time t.
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so friends , you can find the temperature of the object at any given point of time.
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now friends this law is being used widely in the packaging industry
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where food is subject to high temperatures and then it has to be cooled for packaging purposes.
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now lets see how this law works for an ice cream.
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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.
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You need to find out what is the temperature of the ice cream after 5 minutes.
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surely it has increased, but you can find out, by how much,using newton's law of cooling,
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by doing these calculations, and friends, you will come to know
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that the ice cream temperature after 5 minutes is 15.61 degrees Celsius,
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which means the ice cream is going to literally melt in your hands!!
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differential equations are very useful in radioactive decay. lets see how.
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so friends, radioactive elements like uranium, radium , cobolt, etc on being unstable
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spontaneously decay, to give the daughter nucleus and the alpha, beta or gamma rays,depending on the type of decay.
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so we have a law of radioactive decay, which states that the amount of radioactive element that is decaying per unit time
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is actually directly proportional to the total amount of radioactive element present.
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so since the quantity of the element is changing with respect to time,
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we can use differential equations to express this law.
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and friends please note that the negative sign indicates that
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the quantity of the element is actually decreasing as time progresses.
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on solving this differential equation, we will get the relation between
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the quantity of the element and time t,
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so you can find out how much quantity of the element is remaining at any given point of time.
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one use of this equation is to find the half-life of radioactive elements,
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which is nothing but, the time that is required for half of the atoms in the sample to decay.
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so here we have cobolt, which is a radioactive element, and friends, its half life can be found out to be 5.27 years,
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using appropriate values in the equation.
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This half life value is of great importance friends, because it tells us
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how much quantity of the radioactive element is actually present as time progresses.
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And as you can see in this graph, the percentage of cobolt-60
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that is remaining as time progresses is exponentially decreasing, which means,
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Cobolt, which is widely used for cancer treatment has to be replaced regularly in order to be effective,
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and for this, differential equations is very useful.
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the next application is in economic analysis which includes GDP Calculation.
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Friends the change in GDP with respect to time is found to be directly proportional
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to the current GDP of the economy.
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So if x is the GDP of the economy, here is the differential equation that comes in to the picture.
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g is the growth rate, and on solving this differential equation,
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we will get the relation using which we can find the GDP for any given point of time.
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differential equations come very handy in macro economics,
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which is nothing but the study of large economic systems, like that of a country.
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and one such equation friends, is the fokker-planck equation,
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which is a partial differential equation describing time evolution.
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so friends, equations like this, help us understand why some countries
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are poorer than others, what causes recession and how to deal with it.
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the next application is the maxwell's equations.
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friends, this is a set of 4 coupled, partial differential equations
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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
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while the fourth law is the ampere's law.
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so the triangle symbol that you see is the del operator that is used for divergence and curl vector operations.
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So these are four complex laws, but, they are the building blocks
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for all types of electrical, optical, magnetic technologies and innovations ,
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such as, smart phones, mobile phones and computer technology,
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making mri scanners in hospitals,
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electric motors and generators,
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and also to better understand the various phenomena of light.
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the last application we will be discussing is the newton's second law of motion.
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so friends, this law states that the rate of change of momentum of an object
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is directly proportional to the force that is applied on it.
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so since momentum is changing with respect to time, that means we can use differential equations.
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friends there are two scenarios to this.
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if we assume the mass of the object to be constant, then we will get the well-known formula
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that force is equal to mass multiplied by acceleration.
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however, if the mass is variable, that is either it is increasing or decreasing with respect to time,
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here is the differential equation that we will be using.
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the variable mass equation is very useful to study the motion of rockets
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because the mass of a rocket is decreasing with time, as fuel is burnt and propellants are being expelled.
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but lets take a simpler example of pushing a shopping cart.
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so friends, if the shopping cart is empty, we have all experienced
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that it is going to take less force to push it, because, the shopping cart has less mass.
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so less mass less force needed.
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however if the same shopping cart was filled with say, gifts,
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although you would be delighted to push them home,
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it is definitely going to take more force to push it because, now the shopping cart has more mass.
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so more mass, more force needed to push the cart.
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so friends, thanks for watching, and I hope you found the video useful.
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If so, do like and share the video,
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please leave your comments in the comment section below, and do consider subscribing to Enjoy Math.
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So, till we meet again, take care, Bye.
