Series Solutions of Differential Equations Mini-Lecture

[NganaJHone]

I am very happy to see the vidoe from you, hopefully the others also are happy for You A brief introduction to the method of solving differential equations using power series. Intended for students in a first-year calculus sequence.

[Ondelendo]

Steady I Really Like This Video A brief introduction to the method of solving differential equations using power series. Intended for students in a first-year calculus sequence.

[bebeheuy]

Good, I like that you share this video, I wish success always A brief introduction to the method of solving differential equations using power series. Intended for students in a first-year calculus sequence.

[woodng1]

missed my 2hr differential equations class (over slept) and all i have to say is thank you SOOO much lol you don't even know. might not have explained everything i need to know but it helped tons for 10min!!!

[emc031]

Oh my cosh thank you so much, that was so useful! What a legend... Keep up the good work :)

[Magicleprechuan]

Dude, i have an exam tomorrow, and this video saved my life, you are better than my lecturer, thanks.

[burgerking220]

worse explanation ever!

[lovbinfr3sh]

@burgerking220 i don't know why you would say that, he's actually making it really clear; especially considering this is not an easy topic to teach

[bxnaenae]

@burgerking220 you do a better job then

[annie14411]

thankyou awesome video

[worldcupkrazy]

good video! helped me a lot! Thanks a bunch!

[JSSTyger]

The solution is Ae^(-2x)

That took me all of about 5 seconds

[mdiem]

very efficient way to miss the lesson. good for you.

:-/

[kb100]

Why would you go through all that work when the equation is seperable and solvable in your head?

[lespetitsba]

Because he is demonstrating a method.

[haider87]

Thanks!

[rossinole]

This is fantastic. I wish my prof could explain it like this.

[GHOST3R]

sexy

[ModelAnarchist]

where the fuck did the 'a' come from ?

[thefullmonte03]

it is like c, just a constant coefficient

[dewdewil]

thanks a lot for this video..

[CaptainZeep]

MYYYY dIIICkkkk

isss

haaarrrddd

[dilly9986]

really nice video - thanks!

[edwardFking]

ur a dusch hat

[blueshade26]

this video would have interested me a lot more if it had started off with something like, "here is a common equation used in engineering such and such."

[GiantDevilfish]

This is why the rest of the world hates math: what is the significance of all this? To the layman, it's as if one learns rules for the sake of rules. What do these rules have to do with the world? 9 out of 10 mathematicians fail to make this connection for their students.

[Lolibaba25]

You don't the bombs and jet fighters are gonna build themselves?  :D

[philbaileyd021]

well Im an Engineering student and I learned that the rules we learn as Engineers, Architects and so on make it possible to have computers like the mac air, or really anything built today.

[lz42]

differential equations (basically just equations that use calculus operations) are responsible for nearly every breakthrough in classical physics and engineering. i dont think this was posted for the uninitiated but for someone like me who is taking the class of differential equations, its a neat little overview. math doesnt need you to like it. it is the language of the universe, and remains humanity's greatest achievement, without which no others would have been possible.

[philbaileyd021]

I agree with this person.

If I wasnt for these rules of rules, designers wouldnt have the freedom to be unique with what we make possible or advancements in technology and our world. Maybe its students like you whom always ask stupid questions instead of learning these rules. Maybe you should become a teacher to make student like you understand.

[popeye135]

oh this proofs that 2+2=4!

[jasondh22]

Do lefties jack off with their left hands? lol

[dgjiephfurreighR]

I am in precalculus, and I need help with polar equations. My homework is to convert r=theta, from cartesian form from polar (r,theta), to cartesian form (x,y). Can someone explain to me how to convert it? I tried, but it came out incorrect.

[xycun]

i dont know if u still need the answer r=sqrt(x2+y2) and tg(theta)=[sqrt(x2+y2)]/z

[bradkey98765]

didn't get much of that but i am 15...so i'm going to rewiew back to some of the basics and hopefully i'll understand this slightly better when i come back to it.

[WilWil]

very well explained....i hope my calculus prof is as good as you in explaining these difficult concepts

[bute86]

whhy don't you introduce y'=ky and y=ce^kt from the start ?

[svargadk]

You are so great person.

[jpmac1000]

I'm not a mathematician but your proof as to why the coefficients an and bn must be equal seems circular. You say it's not obvious that they are equal, but then subtract them and equate them to the coefficients of the zero sequence. If you can do that why not just equate them in the first place?

[MisterSym]

He expands the zero sequence to show that there are co-efficient terms on either side that must be equal. If he just equates them directly, then you miss the concept of equating co-efficients on both sides.

His way allows you to apply the tool when you have non-zero co-efficients on the right side.

[knmleaguecity]

Great lesson. Are you doing any more?

[goliathlup1]

i would appreciate it if u could say sth about partial diff. equ.

[smartissexy]

Hahahaha, well said.

[surgeonhashmi]

can u also explain the rest of the stuff to me?? as if i get like this now, i may find it alot easier when i actually come to it when i choose to do maths at a higher level

[surgeonhashmi]

dont understand any of that. Then again I'm only a GCSE student. But how can 'y' an unknown variable be equal to a0 + a1x + x(squared) + .... How does any of that make sense ? it doesn't for me anyways. Im only 16, but plz try to explain as best u can

[smartissexy]

That's because y is not a variable, it is an unknown function of x. The solution of a differential equation is a function, and we assume that function can be written just using powers of x (x, x squared, etc).

[metalhead666fucker]

writing out terms in zigma notation would be clearer that method is clear too...

[smartissexy]

Yeah, it's often a matter of personal choice, and I chose this method essentially because thats how it was presented in the course I was TA'ing at the time. The full sigma method can be more powerful, but I find it less obvious.

[metalhead666fucker]

why would you not write summation sign first before write expanded power series? That would be clearer, (for me at least).... but thats not a big deal...

[metalhead666fucker]

thank you buddy I have DE right now... good job

[imperialcoll]

why didnt you write out summation sign and then take it from there? it would have been much clearer

[smartissexy]

I'm not sure what you're asking? Do you mean solving the differential equations leaving them in sigma notation and finding a recurrence relation? If so, it just wasn't the way the method was approached in this particular course, though it is a good one.

[baptistic]

good stuff, man! this is really helpful and informative. I'm an elec engineering student taking a first course in ODE, and this is the single class i find the most challenging; could you recommend any good source of tutorials on the topic?

[darkhedos]

Hello, very good video.

I thought the pace was ideal, I suggest you do not slow it down if you make other videos. This is, after all, a recorded video. If one has trouble with some part, one can watch it again.

[kikililimimi]

Thanks for the answer!

[kikililimimi]

Sorry that this is an amateur question, but whats the difference between an integral and the summation notation. If you apply the function to the rule the results will probably be the same.

[smartissexy]

This is a great question. There are two different applications for summation notation used often in calculus. The one used here is to represent a power series of a function, in which we add up an infinite number of polynomial terms. What's useful, is that often these series converge, which means we can express them as a closed-form function, as we did here.

[smartissexy]

On the other hand, we also use summation notation to represent a Riemann Sum, which is much more closely related to the integral, specifically the definite integral. Here we take a function over some region, and break it into small pieces to approximate the value (often the area) of the function over the entire region.

[smartissexy]

What is remarkable (hence they call it the Fundamental Theorem of Calculus) is that as we take a partition of our region with an infinite number of pieces, the result is exactly the same as the antiderivative (often referred to as integral) of the function, evaluated at the endpoints. Thanks for asking!

[rymen]

im on civil engineering...this helped me for my mid term of Diff.Eq.for enginners. thanks cuz i missed few classes.

[aaron602]

Excellent way to use the forum of YouTube to do real things for real people. This is why I love this site. Keep up the good work, my friend.

[pjozog]

Just thought I'd say that I think its really cool what you're doing. Keep up the good work.

[Devourer09]

I like how people try to tell you a more efficient way of doing things. =) Thanks for posting and as a teaching tip if you ever had the aspiration of becoming a teacher you could maybe go a little slower, or unless you are going to be a college professor I guess not. Good information none the less. =)

[smartissexy]

Thanks for the input. While I do plan to become a college professor, you're absolutely right that I should be going a bit slower. The problem is that Youtube limits one to ten minutes. In the future I plan to get some more sophisticated video equipment and record some full-length lectures, split up into parts.

[boogiedude4321]

If it's too fast watch it again. Why does everything have to be done to accomodate the lowest common denominator? I thought the speed was perfect.

[DiegoHCantor]

A long time since I studied differential equations. Thanks for putting your time into this. It is very helpful to remember this subject.

[danpovey]

I like that you've put this on youTube.

I do have a quibble with your proof that 2 taylor series representing the same function must be equal. I think it is circular. You assume that if sum_n: a_n x^n = sum_n 0 . x^n then a_n = 0 for all n. But this is just a special case of what you are trying to prove. I think it would be better to equate n'th derivatives, evaluated at x=0; each one corresponds to a different a_n.

[smartissexy]

You're absolutely correct, equating the n'th derivatives is definitely a better way of going about it. Thanks for mentioning it.

[777aaa7778]

it seems you haven't got any idea about method of separation of cofficients (don't sure about the naming of the method in English), or Bernule method for more difficult first order differential equations. Solving first order differential equations by expanding a function into McLoran series sounds ridicular. Better show us how to solve PDA's in this way :D

[smartissexy]

I do in fact well understand both methods you speak of. Clearly this is not the most efficient method for solving general first order differential equations, and certainly not one I would use for anything very complicated, as I mentioned above. As stated in the description, this is meant for an introductory calculus course, to synthesize the topics of differential equations, Taylor/MacLaurain series, and recurrence relations, not as the most effective method of solving differential equations.

[lemmiwinks09]

thank you for helping me

[Russoft]

that is a different way of doing it...but I don't think it would be practical for more complicated problems like xy'' + y' + 3y = sin(x), for example. Cool video.

[smartissexy]

You're absolutely correct, its generally used for more simple equations. The motivation is often to define functions by their solutions to differential equations, take for example the Airy and Bessel functions.

[smartissexy]

I find that the process becomes impractical once nonlinear terms are introducted, though something like xy'' + y' +3y= 0 is still fairly straightforward. In fact, to solve the problem you posed, you could combine more traditional methods, such as finding a particular solution by undetermined coefficients, then finding the homogenous solution with series. Thanks for the feedback.

[Russoft]

I just wrote down the first function that came to mind, similiar to one I did on an assignment a while back. I really do think I prefer using laplace transforms whenever possible since it is less book keeping. I am too messy to keep track of all the stupid terms. Doing something like y'' + cos(x)y = h(x), or something, is where I draw the line and say "I have better things to do with my life".

[pitufina86]

so i can use youtube for studying haha, who knew, thanks leftie

[bewparents]

WOW! Smart is sexy!However,

your arm is equally impressive!