Differential equation introduction | First order differential equations | Khan Academy

[@matthewlefaivre8975]

Sal: So I'm assuming you had a go at it... Me: What's a derivative again???

[@kornelkr6639]

😭

[@epicmatter3512]

If you think of a tangent line, which is a line that touches a function at exactly one point, and is really just the limit, both the left side and right side of a secant line, has a rate of change, but is also the rate of change at that exact point. So all your doing is finding the rate of change or more simply rise over run. And these lines are based off of circles, and limits are based off the simple difference quotient, which in itself is just a way to think of rate or change. So all this is, is complex forms of math built off of circles and rate of change(rise over run). And a circle is just based off the idea of a closed loop, and is something people just drew, and rate of change is just the idea about how numbers change over time in a simple linear equation, which is based off of the ideas of numbers and variables, which in itself is based off of the idea of numbers.And a circle is just geometry, which is also based off of numbers. So when you get down to it. ITS ALL JUST NUMBERS! And differential equations are just ways to relate certain functions like derivatives, but actually can relate the world to numbers. ITS ALL JUST NUMBERS!

[@epicmatter3512]

@@kornelkr6639 I got you.

[@angelfalguera9199]

just exactly what I'm experiencing rn

[@rendodge7884]

its also important to understand the anti-derivative

[@d4v0r_x]

you lost me at 0:01

[@walletinsiders]

He says "differential equations are super..." My brain is like "EASY??" ^_^ then he says "Useful" -_- Useful stuff is hard bruh

[@sunitadas8068]

Ikr 😂

[@kinglaserna1533]

Useful? I’ve been learning Calculus since August of 2019 and still have yet to use out side of school

[@leowu1644]

King La Serna When you get a job in STEM them you will.

[@cardinality1975]

@@kinglaserna1533 Surely you've been introduced to some of the applications of calculus? There are a million ways it can be useful, very interesting stuff!!

[@UltimatePrinceofBB]

They are very useful depending on what profession you'd like to take. Calculus is very crucial especially on probability, and there are a lot of professions that requires being good at probability. For example, being an actuary, or having a job at insurance companies

[@charleslucas1784]

The main thing that I like about the Khan Academy is that he makes advanced math that's usually difficult, a lot easier to understand to where anyone can learn and master what he's teaching.

[@MrPerry435]

"I'm assuming you've had a go at it" Quite bold of you to assume that Sal

[@jeffreynorman5221]

As a student learning differential equation but no textbooks provided in the class, these set of videos means everything

[@aominetaiga343]

No offense but how in the world can your professor lecture differential equations without providing texts?

[@josephruvelas9592]

@@aominetaiga343 probably a school with common exams. math department likely has a curriculum and material to teach/ adhere to but i agree, its kind of unheard of. there probably is a textbook tbh lol

[@Podzhagitel]

z-lib

[@momouwu1937]

@@aominetaiga343 It's kind of normal especially if you are doing an exam specific to a particular developing country and the curriculum isn't the same as the Western countries. You just don't know what textbooks to use. Like if you do SAT exams you know you can use SAT textbooks and if you do A-levels, there are A-level textbooks. Still, if you use other textbooks, it can be helpful.

[@Meimingming]

@@aominetaiga343 mine just gives us exercises but we have no theory about it😭

[@user-zg4sn5jo6v]

Finally! a new playlist for differential equations, I've been waiting for a year.

[@fusion9619]

I'm starting my new years resolution a few months early : watch Khan Academy every day.

[@buddychumpalfriendhomiebud9242]

How are you doing with that?

[@fusion9619]

@@buddychumpalfriendhomiebud9242 hmm... started strong, then forgot. Good timing!

[@soraxinmediares4494]

@@fusion9619 good luck man

[@SV42165]

Why?

[@kwasipoho]

Mine this year

[@xogmaster]

Sal: Here's a differential equation. Figure out if one of the solutions in this list of possible solutions is the solution. My professor: Here's a differential equation. Good luck idiot!

[@rocketpsyence]

This is a good refresher so far. Took diff eq ages ago in college and always felt depressed that it was one of those use it or lose it things and wanted to try and get it back.

[@jordanmathewbailey]

your videos are by far the most helpful!

[@seanspartan2023]

Thank you for this. I managed to get all the way thru university and get my math degree without taking a single course in Differential Equations. At the time I didn't care because my work only involved theory, but now I regret it because I'm more interested in applied math and physics...

[@zeroseven51]

Love the way you explain each step Easy to understand Keep up the good work

[@joshuangtseung6948]

I used to watch these vids for GCSEs I'm now doing an undergraduate and I'm still here, who says KZbin is a waste of time smh

[@maxarmstrong885]

A great video; I was highly recommended to watch your vids to improve my maths and I have to say you are great at explaining things. I look forward to the whole series of differential equations being this helpful! Many thanks

[@nightt2040]

I’m currently a sophomore in algebra 2. All I wanted to do was take a sneak peek of what math would be like next. The only part I understood was when he showed the algebraic equation 🙃 what is math

[@katiakominski432]

Hey, not a bad idea, but you took a bit of a leap here :D You might want to look over (limits and) derivative calculus first, and then you'll understand this a bit.

[@ThatWasherNerd]

hahaha I have a calc AP test in 6 days and I don't even know what this is.

[@comprehendnature2404]

In addition to derivatives, you will need integrals and limits to understand differentials. Without them differentials cannot be understood.

[@alexia8511]

CE student here, and I'm also taking a sneak peek of what's coming.. :'>

[@datmo0se992]

WHAT IS MATH?! TEACHER DONT HURT MEEEE DONT HURT MEEE NO MOREEEEE DUN DUNDUN DUNDUN DUH DUH

[@itsboshd]

WHERE DID HE GET e^-3x FROM???????????

[@semiawesomatic6064]

srmorales08 I know this is old, but I ran through it with a general e^ax. You end up with a quadratic who's solutions are 1 and negative three. I'm sure there's a more consistent way in another video you've probably already watched, but whatever.

[@AkashDasYT323]

from Arab

[@elenav1210]

I'm 3 years late but I'm just studying it now. He got it from l²+2l-3=0 Where l² is d²y/dx² l is dy/dx and y is just 1 From there you get -3 and 1 solutions for l.

[@Gypsy_Danger_TMC]

@@elenav1210 Yes just realised you're right. He's used the auxiliary equation

[@tbpom1003]

thanks for the video! I'm taking a differential equation class, and my teacher skips all steps. Now I know what's going on lol

[@jackandonlyjack3268]

Teacher skipping steps? I guess it's time to skip your teacher's class.

[@gm2407]

So you said how a differential equation is made, and you showed proof of how the answer works, but you jumped straight to the answer. Thus confusing the audience. I know you said there are multiple answers, but not knowing how to get there makes it look like magic. Of course I could have missed something. But this felt like asume the answer and find the proof.

[@walktroughman1952]

Many people I know hate differential equations, idk I love them to bits, I find them fun. I thought I'd struggle understanding them cause I had struggled with calculus, but it was a surprisingly smooth ride. They really are pretty fun to deal with. _Unlike the spawn of Satan that is probabilities and statistics._

[@gzddnmz5506]

this guy`s sound is perfect to teach anything :)

[@gimonoko]

When I hear Sal's voice, I automatically listen. I automatically want to learn and understand the lesson. From the years I've spent on khanacademy, he has become my everyday professor/coach when learning. I'm studying electrical engineering right now. Khan academy has its own professor in that department, but when whenever I go back to whatever Sal's teaching, it makes me feel happy to hear his voice.

[@blockbusstar]

This is the second order differential equation

[@hahaha-zi1xf]

what is the difference between the first order and second order?

[@alesolano9081]

@@hahaha-zi1xf the order corresponds to the bigest derivative in the ecuation (if it has a second derivative then its a second order differential ecuation)

[@austinpundit6321]

Sure, but he was just talking about what a differential equation is.

[@dxalrado]

@@austinpundit6321 look at the title and say that again

[@toby9999]

@@dxalrado He's just talking about differential equations. It's an introduction. We don't care about the order yet. The concept already difficult enough without it.

[@jef1ferys]

swear this is second order, not first

[@stevekerp1]

Here's an explanation of differential equations for people who already understand differential equations.

[@amutuhaireeliza8767]

This has been my best introduction to differential equations ever🙏

[@ZPRODGY]

I’m cooked 🫶

[@luvella9795]

Did calc and aced it 5 years ago and can't remember anything. What is the point of my brain anymore.

[@vez3834]

Don't worry, you've learned "how to learn" and that is extremely important. When you've learned something once, it's going to be easier to relearn it when you need it. Start from easier stuff when relearning :)

[@takudzwambereko1046]

I feel dumber since I started college 😂

[@josiahswanson5518]

There exists a set of an infinite number of solutions to an ODE. Reply with an equation that satisfies the above statement .

[@rr.studios]

Mistook this for ASMR for a bit there.

[@MaximQuantum]

I just figured out a great way to study math at school: Watch videos on the topic to get more insight, intuition and curiousity. This is definitely one of the elements that modern educational systems lack, namely recommending great youtube videos!!!!

[@Mathefusions18]

Straight to the point explanation. Good work..

[@duckyoutube6318]

Wow this is amazing.

[@kispangit3912]

Wow. Finally I know now that the solution of diff.eqs are functions. Gee! Good job! A math mystery finally lifts! Not that I can solve any, haha!!

[@pappa6085]

the derivative of e^x is e^x

[@Aisyah-qf2pz]

Thank youuuuuuuuuu.

[@jonaskoelker]

One thing that annoys me about differential equations as they are often presented is that they omit specifying what's a function of what. What helps me keep the eye on the ball is this idea: When solving a number equation, like x^2 = 4, you're really describing the set { x in R | x^2 = 4 }. One description is to list all the elements: { x in R | x^2 = 4 } = {-2, 2}. When solving a differential equation, you're describing a different set, such as the following: { f: R -> R | f'(x) = f(x) for all x in the intersection of the domains of f and f' } or { f: R -> R | f''(x) + 2f'(x) = 3f(x) for all x in the intersection of the domains of f, f' and f'' } and so on. I have a proof using only ideas from my one-semester real analysis course, without Taylor series, that if f' = f (where f: R -> R) then f(x) = f(0) * e_x where e_x := lim (1+x/n)^n as n -> oo. In other words, { f: R -> R | f' = f } is a subset of { f: R -> R | f(x) = f(0) * e_x for all x }. I have only proven that f(x) = f(0) * e_x is a constraint that can be satisfied if f(0) = 0 [implying f(x) = 0 for all x]: I have not proven that non-zero solutions exist, only how they must look _if_ they exist. This is a less precise description of the set { f: R -> R | f' = f } than I would like, but it's a start.

[@kainajones9393]

Too bad you picked "e" as the first example. "e" in itself is a more abstract idea.

[@akshaypainjane2094]

You are doing a great stuff by spreading education! Hats off!

[@toby9999]

After watching a truck load of introductions to differential equations... finally one that makes sense!

[@jinoymathew440]

Thank you very much

[@vivianmader4523]

hi bestie thanks for the help. love ur handwriting.

[@magilas8485]

Probably one of my hardest class during my second year of engineering. I thought I was pretty good with calculus but this one killed me. Dont know why I was so slow. Literally passed this class because of how nonchalant my prof was. Dude was super smart and taught great but dont give af about tests. 10% midterm 12% finals 20% attendance and the rest were just assignments 😂

[@mykey774]

just like me, from the first meeting until now it's almost close to the final exam, I still don't understand this and I'm worried if I don't pass this class lol

[@Sharna0321]

Why is your voice oddly so good...

[@nodiyaa158]

Ikr!

[@patrickoloo241]

that was really useful

[@jinmingtang1545]

I need to review the chain rule of derivative to understand the calculation.

[@rokettojanpu4669]

All it’s about is dealing with derivatives of a composition of functions, i.e. functions wrapped inside other functions. If you’ve got a function of x like f(g(x)) then the derivative is f’(g(x))*g’(x) Example: Derivative of (sinx)^2 = 2(sinx)(cosx)

[@angelakate9922]

I still need to watch this again a couple of times. Because the information just easily flew out of my mind the first time i watched this.

[@mikemustmurder]

I'm pretty sure I'm going to fail this class.

[@emilielee3378]

mikemustmurder pretty sure that i just failed my final exam in three months :')

[@darnelross6357]

did you fail?

[@hman2875]

@@darnelross6357 i guess so

[@islamzerrougui7718]

did u fail ..??

[@angelinebantillo3128]

I'm curious. Did you fail?

[@earllabule4349]

looks so hard, but actually so simple....*Mind blown!!!

[@aslakskailand2589]

what is the software is he using to write?

[@TheRayll]

paint

[@babitashuckin5831]

@@TheRayll no... It's not.

[@purvai4701]

Classic khan academy stuff its iconic right?

[@clyt9636]

Thank you!

[@judgementwithargo2869]

I have no idea what's going on here or how I got here.

[@sultanwiranatakusumah4154]

Thankyou sir

[@Dadalevrai]

Is it order 2 here isn't it ??

[@vasanthr1485]

S

[@hikaru3584]

here for my midterms, pls pray for me.

[@shadmehrmohammadi-io4sw]

same

[@nathaniel4146]

Iam about to take this subject next sem! wish me goodluck! just starting to learn... Imma update you guys how I did after this incoming sem. BTW, Im a CE student from Ph. Best of luck also to those engineering students! we can do this!

[@XDXD15644]

Best of luck. Im gonna take this subject too. IE student from ph.

[@itzyryeji3164]

Bruh i’m grade 12, and planning to take ece, we have De this sem uwuuu from philippines too

[@snullahc13]

so, how is the course? I also consider taking DE next semester...

[@Engr.S-Jay29]

Im an ME student from Philippines. I have my Calculus 1 this semester.

[@waltercatapang7159]

Same wd you bruh, God Bless Us

[@projectgreengamer]

My college algebra teacher has us trying to learn this almost right at the start of the semester. I signed up for algebra.. why are we starting with calculus :(

[@ulicadluga]

Why are Math teachers so bad at explaining things? Not you, Sir. You seem intent on conveying something. I was literally abused by my math teachers. Not a word about what "this " might be good for in the "real world". Bad teachers are "baaaaad news"!

[@d00kie666]

Oop my bad I'm only here because I'm a huge SIMP for this man's voice... Likeeeeee omg

[@theomiado7954]

this is going to be a bumpy ride

[@thatnibba2551]

This is helpful

[@AmbientsonarVA]

I’m thinking about doing chemistry as a minor in college next year. I heard that differential equations are a must for the fundamentals.

[@ricardoalves9605]

What's the point? You gave 2 examples for functions that would work, but didn't actually explain how we get to those said functions in the first place

[@vez3834]

The point is to show some examples, and to showcase how taking derivatives and adding them isn't anything too scary. It's an introduction, not a full explanation of the subject.

[@elmoreglidingclub3030]

So, newbie question, how/why did e just suddenly and magically appear??

[@jakeroly2593]

My professor teaches for his first time ever, so his teaching is kinda awful and inefficient that I had to rely on teaching differential equations to myself.

[@structuralanalysis2790]

Where are you at?

[@Shuforces1]

A question, are there any lectures/materials that I should look at before I look at this? At 4:08 I have no idea what he is asking me and how I am supposed to do it.

[@AWESOMEDAVIDPOWER]

You should look into derivatives. He y' is the first order derivative of y, and y'' is the second order derivative of y. y in this case is e^(-3x). To learn how to find the first and second order derivative of e^(-3x) make sure that you learn the chain rule part of derivatives as well :)

[@UnitedPebbles]

What you do not get? Oh where does he get the y = e^(3x) from? He forgot to said: it is given? And then to said oh the solution is equals to itself is not correct?

[@nwjt]

what software and hardware are you using to write with? it looks nice

[@vsainscorp.4393]

please.. Add Indonesian subtitles...

[@abdulrahmanm.bashir4878]

he just said "that is the start"...hmmm i wounder how the end will look like!

[@talalafzaal7906]

Didn't understand a single thing. Waste of time.

[@ignantxxxninja]

very nice introduction.. I'm reading the preface of the textbook before the semester starts, and it seems like this is going to make sense.

[@transgalactic]

Yes it checks out, but how did you arrive to the solutions?

[@AbiM-G]

you divide the digit that has the y or x by that number and then simplify both sides

[@AbiM-G]

you understand 5x means (5× x)

[@AbiM-G]

eg 3z=6 so 3 3 then simplify ,then you will be left with z=2 so that means 3×z which is 2 is 6

[@Fredfredfredfredfredfredfred]

Could someone recommend a path from algebra to here? I am good at algebra, like in school I could graph in my head. Not kidding. But I want to understand this and I’m a wee bit ill equipped

[@leonardebron6032]

Why bro sound like DJ Vlad?😂

[@rickybosephus2036]

He didn't explain it. FAIL

[@Johan-st4rv]

What about f(x) = 0

[@marianmortada8978]

where general solution of diff equetions come from

[@rokettojanpu4669]

marian mortada A differential equations class will teach you the derivations for all sorts of methods to solving different types of differential equations. Most DEs cannot be solved by simple separation of variables as taught in your first calculus class.

[@remyendaya7103]

How to donate using GCASH

[@Casanova102986]

where did the y1(x)=e^-3x come from ?

[@PHXKHEEDGAMING]

I can’t find the other tutorials

[@drmksu38]

thank you Mr kahan .. i cant do anything without your amazing channel ♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️

[@Sibearian_]

i don't learn anything :(

[@letoheeo3970]

Why

[@Sibearian_]

@@letoheeo3970 why not?

[@letoheeo3970]

@@Sibearian_ what degree you had or have u been studying?

[@ikesteroma]

Your handwriting is excellent.

[@marianmortada8978]

why all genral solutions of diff equestion contain e to the power some thing interms of x

[@rokettojanpu4669]

This isn’t true. Exponential functions just happen to be the type of solutions to the DE examples Sal gave in this video, which are second order constant coefficient DEs.

[@carlkpsplucky5554]

Is this algebra or calculus

[@esang2667]

BAKIT AKO NAG ENGINEERING? 😭😭😭😭

[@xAZN-UAE_Legendx]

This doesn't make any sense. Where did all the "e's" come from? Now you're telling me differential equations are done without a calculator? How do you introduce a new letter into a formula that never included that letter to begin with?! When the formulas consist of y's and 2. Ok. You've done the first equation but how about the other 2?! Please explain! This is more complicated than escaping a typhoon.

[@Colonies_Dev]

How can you say e^-3x and apply the power rule and then ^-3x stays the same ;-;

[@Colonies_Dev]

Oh it's bc the derivative of e^x is e^x and 3 is derivative of 3x

[@QuackDocElias]

man, it never really clicked til you said it but of course the solution would be a function... just like for x's it'd be numbers.

[@hashemhassani1792]

Please I need help in this problem Y"*Y^2=-K^2 WHEN :t=0,Y=A,Y'=B,Y"=C

[@jasdrawssometimes4660]

i wish there was highschool college where you would take college courses on a highschool schedule. so like an hour- 2hours a day, everyday, for a year. i would be so good at differential equations. i wish i had the time to love math the way i did in highschool

[@gregsuarez5205]

Thank you. This is great and helps me understand . Can you let me know what tool you are using to draw and illustrate in your presentation?

[@balling220]

Engineering brought everybody here Me: Economics brought me here😮

[@yusafkhan2000]

I hate maths 😭

[@sethbaldwin131]

But you've come this far???

[@johnthegamerman404]

Power series expansion method is my favorite way to solve a differential equation

[@ThePeterDislikeShow]

Sal, you operate on donations?! Your channel ought to be making a fortune!

[@JuliaLisa-c3n]

Brown Robert Hernandez Sandra Lewis Thomas

[@Marcus001]

I could’ve taken this in HS…