Calculus - Differential Equation Example

[@hmdz150]

This is so helpful, thanks a lot Derek Owens. Unfortunatelly most universities teach differential equations never mentioning where they actually come from! And how helpful they are solving everyday problems in physics and other fields of science.

[@derekowens]

Thanks! And you are absolutely correct. At most schools, it is the math professors teaching DE, and it is probably mostly theory. If you are studying where they come from and how they are used, then you are probably studying physics or engineering, and the physics and engineering professors usually assume that you already know DE!

[@hmdz150]

Totally agree, I studied EE and it wasn't until analysing second order RLC circuits that I realized how amazing and powerful DE were! I think similar tutorials on PDE and how they are formed (like wave equation) by you would be awesome:)

[@hg2.]

Yes. Next up: PDEs. :)

[@hg2.]

BRAVO! I've been mystified/mesmerized by "differential equations " all my life! Here they are! Thank you so much Derek!

[@ItzAnArt]

I'm a week away from finishing my Diff EQ class and this actually helped me understand what this class actually does lol. One more way to explain to others of how I'd use this class in real life. Good video

[@davidkippy101]

I'm just discovering how powerful differential equations are in describing reality. All you have at the start is a relation between a value and it's rate of change, and in the end you can map an entire function of time of that value. It's crazy.

[@ShivamVerma-gq2sm]

Sir I was spellbound... Thanks for uploading such an educative video... If people like u continue to come up opening their minds for the learners then I m sure no would be litrated but everyone would be educated :D

[@derekowens]

Exactly. The reality will only match the theory within certain limits, and discerning those limits in various particular situations is a very valuable skill.

[@priced80]

Yes, but depends on what you are doing. You would be surprised though. Physicists love to keep things as simple as possible to reduce redundant complexity and make calculations easier etc, so sometimes even really difficult problems have elegant solutions (like e=mc^2)

[@derekowens]

Yes, a lot of people say that. Our videos do look similar, although behind the scenes we are doing different work.

[@Asahel717]

Great work, Derek. Thank you. I subscribed to your channel on the basis of this video. Looking forward to learning more from you.

[@143143akash]

Boooom !!! Mind blown !! Great example very helpful . Awesome work 👏👏

[@VFizz]

Excellent example Sir!! Considering that I am still fairly new at this, you really do make it easy and logical! Thanks again!

[@imegatrone]

after i watched this video, my insight is very open because the video is very good to give information An example of a differential equation with an exponential function as a solution. A fairly standard calculus problem.

[@thenatmann1]

mid 2020 and still relevant! I wish I could have had you as one of my professors!

[@Geodendronitrian]

you stupid man

[@Alaska-mk4ok]

Mid 2021 and still relevant

[@JacobTJ1]

this is great thank you, i searched for a practical application of differential equation and this was just perfect.

[@osvaldodossantos9354]

Dear Professor: can you Tell me please why the plane never line up on the road? Many thanks!

[@Virsconte]

This is just my guess, since I am still very much a student of math. I think it behaves this way because theta gets closer and closer to 0 degrees, meaning that after a long time, the rear wheels are mostly pointed forward, and only creep slowly towards the line. Also, more mathematically, the e^(-x) has an asymptote at zero. If you plugged in y=0, you would get a false statement, because there is no input that can give you exactly zero.

[@kingjoe8788]

why wouldnt the plane get 100% in line? why couldnt we replace y=0?

[@JrmcStudio]

yea try plugging in y=0 in the calculator, you will see that it will be undefined.

[@adminassistant4409]

Ma main man, Derek! Can always count on you to get some solid knowledge. Thanks!

[@josefrancis3714]

Lovely and competent lecture! Thanks.

[@hmdz150]

@virsconte The meaning of infinity is different in physics compared to math. For many physical problems a distance of few yards equals infinity.

[@1hassan1100]

Mr, owens, would you get into partial differential equations, because what i could'nt preserve in so many years looking for understanding , i understood from you spountiniously

[@sajadhussain761]

These videos really act as path finders for people like us. We learn a lot from ur videos...thanks sir for such a marvelous job.. Sir kindly tell me on which application u record videos. Ur videos r so pretty and clean. Regards

[@derekowens]

Thanks, Sajad. There is nothing tricky about creating these videos. I'm just doing screen capture while writing on a tablet in a drawing program. I use Camtasia for the screen capture, although I believe there are also some free software choices that will work well. I have used a variety of tablets, most recently the ones from Huion. Hope that helps!

[@don_miko]

thanks for the helpful video. why is it that the plane can never be precisely alligned with the line? and can we show that mathematically?

[@danielscott853]

That is the nature of a log.

[@pathfinder44ltd]

If you observe an actual B767 making that turn, the path of the main gear tires will ARC towards the line with the RIGHT TIRE tracking on a WIDER ARC THAN THE LEFT. Also the distance between the tires are not considered within this calculation, ALTHOUGH the difference between the calculated solution & the actual 'event' is small!

[@Agntjpa]

This is similar to the steady state response we get during studying transients in Control sytems

[@blackhole2345]

Very well done, I remember when I was in college I somehow hated these... Now they make so much more sense!

[@mommonyratana4685]

at 5:10 why did you integrate it?

[@jmjmjmjmjmjmjmjmjmjmjm]

It's actually 55 metres long but thanks for the awesome video, your channel is very well done!

[@danielscott853]

What threshold did you use?

[@georgemardari6568]

So i get that x = vt, and when you differentiate that, dx = vdt since v is a constant. but when you do dy = v dt sin theta, you treat sin theta as a constant right? shoudlnt you differentiate sin theta, since it's changing? like cos theta dtheta. Please help my train of thought, I'm a beginner to Calculus.

[@wesleyblack8302]

why can it never completely line up?

[@alexistorres8647]

They will line up, but it is obvious that once the plain in on ground, the plain and the ground are completely lined up. The question is, how long will it take the plain to be almost but not exactly lined up with the ground right before the plane comes into contact with the ground.

[@radzelimohdramli4360]

What will happen if y=0 or y approaching 0?

[@wesleyblack8302]

its undefined. if you solve the equation for t, you have: t= (L*ln(y_0/y))/v but in order for y=y_0, they both have to equal 0 because e^x cannot equal 0 for a real value 'x' meaning [y=y_0=0], but that means that our ratio in our 'ln' is 0/0 and ln[0/0] is undefined.

[@adelalkhayat6783]

At the moment when the front and back wheels of the plane get aligned on the main runway (horizontally), Y must be zero. As such, there will be a problem solving the integrated equation: Y = C e^( -vt/L).

[@benjaminkarazi968]

Your logic=Neverland planes! Study physic.

[@furuyaman]

Thank you! Super clear!

[@bobbyyankey5967]

Good illusrtation of real application of diff. equation

[@fazalnajam]

That's a great example. btw which software you used for writing? The fonts are so clean

[@L6FT]

How is: dy=dx*sin(@) ?

[@alkankondo89]

I was wondering that same thing! Intuitively, I would expect dy = dx*tan(@) because then we could write dy/dx = tan(@), which according to the picture should be correct.

[@thelastcipher9135]

dy is the infinitesimal vertical change in position. Since dy is opposite to theta and the hypotenuse of the triangle with angle theta is |-v|dt, then sin(theta)=dy/v*dt. Hence, dy=v*dt sin(theta). We put a negative sign to indicate direction so dy=-v*dt*sin(theta).

[@barrjohnm]

Great video, thanks!

[@WCGwkf]

I don't know integrals quite yet but this whole thing seems fairly simple and it's all familiar to me. I'm assuming this is a really simple example and they get much much more complicated, or what?

[@thanyaniinnocent940]

I love this video 🇿🇦♥️

[@putteneersjoris]

very nice explained!

[@thelastcipher9135]

For people wondering the derivation of the formula for dy, here is an explanation: dy is the infinitesimal vertical change in position. Since dy is opposite to theta and the hypotenuse of the triangle, with angle theta, is |-v|dt, then sin(theta)=dy/v*dt. Hence, dy=v*dt sin(theta). We put a negative sign to indicate direction so dy=-v*dt*sin(theta).

[@raguls4009]

it's very useful to my modeling

[@madisonelectronic]

Once on the line and tracking it how fast (at a given speed) does the nosewheel have to turn to stay on track? hehhehhehhehheh

[@TheBriansle]

that was beautiful. thank you.

[@SuperSaverPlaysSPG]

Great videos on calculus!!! I'm taking calculus for my economics degree in UNR and boy...the class is sooo superficial!! I finished a full semester's assignments in the first month and wondering...this can't be it...Calculus must be more than this. Your videos showed me Yes...there is MUCH MUCH more to it than what my school teaches... Now I understand why soo many new graduates are utterly unprepared.

[@Yashvermarox]

THIS WAS REALLY AWESOME ........... BUT I HAVE A DOUBT #Derek Owens in which direction is the plane moving initially

[@pattystomper1]

I thought you would be tracking the front wheels along the horizontal X axis, yet you kept referring to the hypotenuse L. In other words, instead of calculating using sine, wouldn't you use tangent?

[@md.shakilsheikh1744]

Thank you so much from Bangladesh.

[@diegeeleel]

lost me at 'dy = -vdt sin theta'. 'dy' being the opposite side and vdt being adjacent to theta should it not be dy = vdt tan theta? as tan theta equals the opposite over the adjacent side and therefore tan theta multiplied by the adjacent side equals the opposite side?

[@thginkkrad6554]

+tbone vdt is the hypotenuse, not adjacent side

[@Kevin-rf5yp]

+Thgink Krad the problem is how do you get dy = dx•sine@ ? if you started from y = x•sine@. For sure we have sine@ = L / y . L is constant and y is variable. and here @ as well variable. But I do not agree you made equation jumped to dy = dx•sine@ . and you got dy=vdt•sine@. show me how come?

[@Kevin-rf5yp]

+want to know DE sorry.. there is mistyping.. sine@ = y / L .

[@Kevin-rf5yp]

+want to know DE And I totally agree with tone for '-'. the ' - ' is opposite direction and it does not mean getting smaller..

[@Kevin-rf5yp]

+Thgink Krad also as we know the speed v=5m/s. Do you really think the speed of plane is just 5 m/s ? as you said x direction is the direction of flying. so plane's speed is 5 m/s which mean 5×3600 m/ h so 18 km/h. this plane is so slow. do you know marathon record? they can run 42.195km at around in 2 hours. so this plane is slower then human...ㅡㅡ;

[@mohfa1806]

nice video , but can anyone tell me why the direction of the plane is straight oblique on dx?

[@TheGamecheater100]

nicely explained

[@Y10Q]

Did you ever get your friend to test this? It doesn't seem accurate to me, even though, theoretically it is correct.

[@snowyearound]

That does seem like a long time for y to reach .1m. Moving at 5m/s you'd think it would be faster

[@XPLAlN]

Although it does feel like a long time, if you do the sums for reducing y from 10m to 1m (instead of 0.1m) it takes only half the time. Such is the nature of the problem.

[@WCGwkf]

.... the font is writing with your hand..?

[@Kevin-rf5yp]

If I follow this solution. This plane's speed is just 5 m/s and 18km/s. so very slow and slower than marathoner. The problem is x direction. x is not a distance of flight of this plane. the distance of x axis when this plane is moving. and 5 m/s is the speed of shade of this plane reflected from x axis.

[@Kevin-rf5yp]

sorry mistyping.. 18km/h.

[@honestman276]

Thanks. From Bangladesh.

[@priced80]

Great video, thanks. This is assuming no noise in the system right? Of course in practice after y had reached < a few mm the wheel would keep crossing the line due to imperfections in the wheels, run way, wind, etc.

[@SpellingBeeC]

this is really helpful

[@TheStraightThinker]

It's the natural logarithm base : e = 2.718281828459...... occurs a lot in calculus.

[@mpbgd]

excellent

[@derekowens]

That's exactly right!

[@lachuck2476]

Okay, my question is...Doesn't the original location of Y (Where the wheels are ) move diagonally which is a longer distance than the vertical for value for Y?

[@RajeshKumar-ch4rv]

Easy and impressive

[@billydagenham]

i didnt think george bush would know differential equations

[@Kevin-rf5yp]

This solution has problem. How dy = dx•sine@? just skip the proof.

[@ananthmadan3611]

dx=-vdt sin@

[@wesleyblack8302]

'dy' is the vertical projection of 'dx' in this model. dx is a radius/hypotenuse, so the vertical comp is given by dx * sin(theta) negative in this case, but since dx is differential instead of being y, it is dy

[@JB-mo1sw]

I'm permanently cross-eyed 😮

[@amandaeads1993]

thanks manny dejesus washington il

[@Zander101084]

I don't think you sound like George Bush. Great video though! Thanks! No need to reply (you're probably busy).

[@Geodendronitrian]

this video is cool

[@iamwangine8752]

hello Derek, cool explanation. You can use part of your video in my channel? . Subscribed and thanks

[@TheStraightThinker]

Anything to the zero is 1, except for zero itself and infinity.

[@digitisgood]

thanks a lot... :)

[@55chh]

Isn't it actually more complicated than that? The path of the 767 doesn't follow the slope of the triangle, if it did the nosewheel would go off the line. The rear wheels do not take a straight path (y) to the line, they must follow a curve which slowly converges on the x axis line.

[@mohanadalnuaimi3238]

when y--> 0 means theta -->0 , theta have to be considered as a variable value, sin(d(theta))=(y-dy)/L ...and go from there.......

[@navinlal9161]

Why there are sinϴ, why not tan or any other

[@zabthecar]

LOL ! At the terrorist comments ! But great stuff ! Thankyou. =)

[@dravidr007]

thumbnail looks like 9/11 calcs!

[@nicholasmerino9608]

Lol!

[@Netlocd]

LMFAO

[@emilynunez5549]

I hate math

[@draganarbutina8835]

Ne razumem se u ovoo. Godine i da je

[@khaledalhorani7965]

Just one single word: "WOW"

[@8day1989]

Sorry poor stupid man for asking this, but how did he knew that he must use sine function? I.e., how it relates to the shown example?

[@CharlesFrederick]

He pretty much broke the polar function into multiple Cartesian components. The x would equal to cos(theta) as well. It's how right triangles work.

[@rahulupadhyay3570]

8thDay sin(theta)=y/L. thus y=L*sin(theta)

[@Luvthewayumove]

a plane has two sets of rear wheels.

[@academiainglesgarantizadom1777]

too many shortcuts.. alot of poets study math as required at college and lose the idea that its all about y relating to x-- we use deriviand tives or integrals which are shortcuts..we need to show that it all boils down to y to x. a line is easy for a poet to understand and the curve is seen as a line in calculus. constantly remind the poets that we use shortcuts.. the derivitive is not voodoo. its a shortcut.. and the answer must be practical.. saving money by 100 dollars more per day..how many days to get 5000 dollars. or how many cakes to bake ..too many is to lose money.. you can show it to a poet.. somany students hate math cause of the shortcuts... they see it as magic..the logic is forgotten by nerdy teachers , my best calc teacher was an gym teacher like me and he taught me in simple terms.. the above problem i see perfectly.