Power Series Solutions of Differential Equations

[@mr3zzoz]

this guy will go straight to the heaven

[@parkerflop]

Godwilling

[@sasfa1]

so is it a highway heading north bound or like subway... maybe a interstate?

[@kaizhang680]

Patrick, I just want to take my time to thank you for all the great videos on your channel. I am a struggling engineer but thanks to your videos, i've been passing all my math classes with ease. I really wish that you have more advanced contents or covered more topics beyond simple calculus such as fluid mechanics or statistic analysis so that i can breeze through all my courses. It literally takes me 5min to fully cover a topic compared to 2-3 hours by looking through my textbook. Anyway keep doing what you do.

[@Magnawulf]

For anyone wondering why it's not Cn = Co/(n+1)! Look at his equations. For example he determined that C3 = Co/3! which is indeed Cn = Co/n!. He also determined that C4 = Co/4! which follows Cn = Co/n!

[@bangargaurav1636]

Magnawulf thanks for explaining

[@TheRevival88]

Here starting level of numeral too matters

[@Vyantri]

When you say C3, that subscript 3 does not represent just n, but n+1. So I think that the final result can be written as C subscript n+1 equals C sub naught over (n+1)! and I think that this is equivalent to Cn=C0/n! You can replace the n with anything because they are the same thing in both the left and right hand side of the equation.

[@tonystank1309]

My man, @PatrickJMT is helping a lot of people.... Especially during their exams!!! Thanks man.

[@magaliebravo6679]

I'm pretty sure its suppose to be c/(n+1)! either way, thanks for uploading PatricJMT!!! I've been following you since my algebra days.

[@vaultvon2126]

Same here. He is the master. from the very start to ....

[@sayanmondal4570]

Magalie Bravo Actually for n=m-1 we get the value of Cm = Co/(m!)

[@johnnysv7646]

It would have been better to "Re-index" or change the name of the variable "n" to "k" to avoid confusion because to determine the pattern you look at the consecutive Cn values. So for C1=C0/1! , C2=C0/2! , C3=C0/3! we can clearly see that the pattern is in fact Cn= C0/n! or Ck=C0/k! which is how I write it to avoid confusion.

[@Peter_1986]

PatrickJMT, Professor Leonard, Khan Academy, blackpenredpen and Michel Van Biezen are my favourite applied science channels.

[@MichaelSmith-rh9ok]

@@Peter_1986 The organic Chemistry tutor is really good too!

[@hampsternips]

my lecturer is a potato, Patrick you are awesome!

[@mathadventuress]

Hahaha

[@shaquillchole8042]

😂😂😂wow

[@moosaabbas8310]

Lk1hlIU

[@Ammaray]

Would you be willing to post a couple more examples of Power Series solutions of DE with initial conditions, possibly in a term-by-term method? Thanks for all you have done. You have helped me so much already! I have been learning from your videos for a couple of years now.

[@heilgid92]

the man, the myth, the legend... patrickjmt

[@syedubaid4116]

The man, the math , the legend

[@heilgid92]

@@syedubaid4116 wow! A reply to my 7 year old comment!

[@syedubaid4116]

@@heilgid92 lol, it's KZbin's new algorithm which brings old comments up

[@heilgid92]

@@syedubaid4116awesome. Why are you learning differential equations? 7 years ago I was in college. Now I've been working as an engineer for over 5 years now!

[@syedubaid4116]

@@heilgid92 good question, I completed all my engineering courses 6 years back apart from differential, couldn't get my degree ... That's why I'm studying differential... Me too I'm an electrical engineering on job since the past 6 years 😂

[@MrSexyJem]

patrick, for the people who are going to have trouble in the near future, i really suggest that u put up another example one with x's and y's because those question really needs a next level of understanding. For example y'' - xy' - y= 0 really suggest you put up on any of these type of equations. it would really help us out alot.

[@Zeeblog]

Thanks, this really helped, your teaching method is really clear and concise. I also like the way you make your summation signs, haha.

[@mosulicwele1824]

I passed the maths because of your efforts, may all the blessings be upon you. Thanks!!! You always make things easier so that we can understand.

[@virginialikesyou]

patrickJMT you have helped me from Calculus I all the way through Differential Equations. I LOVE YOU!!

[@JustinZymbaluk]

Thanks a lot Patrick, you've been a real life saver my entire college career. I also just want to say, I love the way you write your sigmas, they look so cool and sleek!

[@Arwokid]

because (if im not wrong) ...a power series has to start with x^0 (

[@patrickjmt]

differential equations - usually taken after calc 2 or 3

[@bobyoco]

@patrickJMT I second the idea of another video even though I have my test soon. This was the hardest for me to learn/understand. As always, you're a lifesaver Pat!

[@jossell911]

God Bless you patrickJMT. Just in time for my Diff Eq Quiz today!

[@patrickjmt]

it is: QuickCam Orbit AF

[@damienjamieson7577]

Thanks Patrick for a clear reliable lesson.

[@pamodakoggala]

patrickJMT thank you very much for uploading this video. It was really helpful

[@tokkeekkot6874]

🎉

[@JSSTyger]

@tfreed20 I can clear up your question. When he had Cn+1 = Cn/(n+1), the equation was in terms of Cn and not of C0. That is why the multipliers were not the same. Plugging in n = 1, you get C2 = C1/(2) = C0/2. It actually is the same.

[@TazerBlaze]

When you differentiate a summation, imagine differentiating every term in the summation. He changed it because if not then, you would start at n=0 for all terms in the summation so the 1st term would be C0*x^-1 which would not be the derivative of ANY of the terms so it would not be in the summation, that's how it has to be rewritten (:

[@Ammaray]

Question: What does it mean if an initial condition is given, such as in the following example: df/dt = f + t init. cond.: f(0) = 1. Thanks for all the other vids! You are a life saver.

[@patrickalle]

Hey Pat. At the end of your video you're basically saying "oh, well this is basically e^x." Which brought Laplace Transforms to mind. Where you have to recognize what whatever the thing is and plug and chug. Are these related or just coincidences?

[@Isou3008]

You are always on point Patrick , thank you so much.

[@patrickjmt]

you are very welcome

[@Syruscleat]

Wow! You made my hat spin with all that algebra! As always, my hat goes off to the AMG! American Math Guru that is!

[@aubreehuntley5588]

thank you your video is so much clearer than others!! Clutch for my calc 2 final❤️❤️❤️

[@kagayakuangel5828]

I am laughing when he said at the end, “hopefully this looks familiar to you” 😂 and I’m just like, “he means that’s e^x right?” Lol and lol yeah. I hope it looks familiar to me on that final exam next week LMFAOOOOO!!!!!!!!!!

[@VelocityCCX]

Thank-you Patrick, this is my first of I hope many video's I will watch.

[@QuackersForMath]

Really great explanation - something my university failed to achieve. Thanks mate :)

[@geromegacrama284]

Wow ,amazing ,,thanks Teacher 😁😁🙏..God bless you 🙏

[@DaveDruska]

Thanks for this, post that harder question! In your next video can you explain how to find the first few nonzero terms "about x=0"?

[@JSSTyger]

I knew the answer right when I looked at it. :) I actually did not learn power series methods for solving diff eqs in college. I taught it to myself afterward in my spare time.

[@patrickjmt]

thanks for the tip! i will do this

[@PatWizard]

bro thanking his own video. I can't even blame him, its that good

[@vbballmaster]

its because the formula is kind of a shorthand to avoid doing the calculations, its not Cn, but C(n+1) originally, so n=2 gives C3 = Co/3!, which follows with Cn= Co/n!

[@thies2us]

Patrick and others, do we always have to assume a solution of this kind? are there any other solutions to assume? thanks

[@shaygahweh]

I got a 100% on the test thanks to you Patrick =) Last calc course. WOOHOO! =D

[@benclark5747]

why isn't it no cn=c0/(n+1)!

[@socratiat_0]

The question at any minute of the video .. so I can help

[@maddywellsgarske799]

Ben Clark This is because he is NOT using the same (n) as he was using when plugging in values to find the coefficients. Look again at 8:35. When he was plugging in n=3 into the equation he was getting C4=Co/(4!) . The key point here is noticing the 4's! You can also look at the rest of the examples before that C3=Co/3!, C2=C0/(2!) and so on. Therefore he can say Cn=Co/(n!), you could also say C(n+1)=Co/(n+1)! if you wanted to use the same n in both ideas. Try plugging in (n) into your equation. Lets try n=1. With your equation C1=Co/(2!), is this correct? No, the answer should be C1=Co/1! or C1=Co. So you would either have to change your equation to C(n+1)=Co/(n+1)! or Cn=Co/(n!) in order for it to fit the pattern. Both are correct! Yours was a mixture of both. I hope that helps!

[@cpt_shipard9698]

@@maddywellsgarske799 Thx for pointing that out.

[@anzatzi]

wow--that was awesome--was jest reading the same example in a book with no results--thanks

[@DoggoWillink]

YES!! Patrick my exam was cancelled today because the "courier", if you will, never showed up with the exams haha. I might have a chance now. He said it was a first, just left now; too happy.

[@Thexder0]

@liquidstl Yes, you first shift indices to match the exponents in x. If after that, the sums start at different points, **do not** shift -- just write separately the few indices which are not in all the sums. That said, great video!

[@da1booger13]

This is why I love this place. I'm sure to learn math and get a good laugh at the same time.

[@johnnysv7646]

His pattern is correct Cn= C0/n!. It would have been better to "Re-index" or change the name of the variable "n" to "k" to avoid confusion because to determine the pattern you look at the consecutive Cn values. So for C1=C0/1! , C2=C0/2! , C3=C0/3! we can clearly see that the pattern is in fact Cn= C0/n! or Ck=C0/k! which is how I write it to avoid confusion. I hope this cleared up any confusion

[@coltennabers634]

if you go to TTU don't get Arne Ledet for Diffeq - just a heads up.

[@iDanielSalmeron]

Wreck EM wreck EM wreck EM

[@canyuksek4224]

You are a great tutor, thanks!

[@td-po5ik]

Patrick ! this question was on my exam. Thank you so much !

[@reclusivewanker4119]

that was really helpful but in my humble opinion you should do more complicated ones.

[@rabeeaatif1984]

YOU ARE AMAZING!!!!!!!

[@abguitar99]

Thank you patrickJMT :) Really helpful.

[@liquidstl]

I'm taking an engineering math class this fall so I'm watching some of these videos as review. One thing I was never too sure of is. When shifting the index; is the point to have the index start at the same point for each series, or to have the same exponent so that you can factor out the Cn's and equate coefficients to solve for the recurrence relation? Also, Its slipping my mind at the moment, but other than inspection, what other way is there to solve this ODE?

[@javierrubio1889]

Thank you for your videos. This example is too obvious ( y' - y = 0, y' = y,....what's the proper function of the derivative, Dy=y ?....it is y = c exp x ). But it's nice to see that the method works! . JBcn

[@carbon273]

how did you get C_n= C_0/n! if for example at n=3: C_4=C_0/4! ???? if n=3 and the subscripts are 4 shouldn't it be n+1 instead of n for the generalized form?

[@patrickjmt]

no, there is no mistake!

[@markkennedy9767]

At 3:00 isn't the goal of shifting index to make the powers on the x the same (to allow us to add both sums) and not necessarily to have both sums to go from zero to infinity as you say in the video.

[@zurchei]

that was helpfull indeed , and will be glad if u could put more vedios on differentials equations..they are realy gıving me headaches..

[@ibrahimhelmi2335]

Thank you very much,,looking forward to see example 2 :)

[@DaveDruska]

You should turn off the auto brightness/contrast so the whole video doesn't flash brightness when your hand goes into view. Thanks!

[@ujueije5762]

Brilliant, really brilliant for this excellent video. Keep on the good job

[@kellyblakmoffat]

my secrete tutor...you have been so helpful just from first year now am in my third year EEE

[@patrickjmt]

time to go be a donor at Patreon :)

[@ovsick]

Can you please put a video on how you get n+1 =n , lagging. When you have coefficients (c1, c2, c3...) equal to zero. How you do the lagging, and how you know how many times you need iterate the Sum. Thanks

[@Peter_1986]

My Linear Analysis class spent a whole lecture going through this stuff. I skipped that lecture and watched this video instead, lmao.

[@kimkieu3010]

Very helpful, thanks. But my DE homework is so much harder than this :/

[@yorkeR177]

@evelelamayo I don't believe so. If you check, he arrived at the condition: C_(n+1) = (C_n)/(n+1). He then wrote his answer in terms of C_n, so (C_0)/(n!) is the correct answer. This is actually a famous example because y = ce^x (constant c) is the only function where y = y' (I think).

[@wiselinkmarque3947]

all the problems I find online are too simple compared to my differentials... I can't figure this stuff out. went to two college classes, tutors don't have a clue.... this sucks.

[@magaliebravo6679]

wiselink thats when its time to meet with the professor

[@wiselinkmarque3947]

Magalie Bravo That's all fine when the professor has time but my professor had grandkids to attend to a lot so he wasn't around much. I passed the class anyways though. I just don't see how it is going to help me with finding a job later on.

[@maddywellsgarske799]

Magalie Bravo This is a reply to your earlier post and the reason behind why his equation is actually correct. I thought it wasn't at first just like you did. Here is my explanation. The n he is using in the end isnt exactly the same n as he was using before. Look again at 8:35. When he was plugging in n=3 into the equation he was getting C4=Co/(4!) . The key point here is noticing the 4's! You can also look at the rest of the examples before that C3=Co/3!, C2=C0/(2!) and so on. Therefore he can say Cn=Co/(n!), you could also say C(n+1)=Co/(n+1)! if you wanted to use the same n in both ideas. Try plugging in (n) into your equation. Lets try n=1. With your equation C1=Co/(2!), is this correct? No, the answer should be C1=Co/1! or C1=Co. So you would either have to change your equation to C(n+1)=Co/(n+1)! or Cn=Co/(n!) in order for it to fit the pattern. Yours was a mixture of both. I hope that helps!

[@ayuparpe1580]

how are u r doing ENGINEER WISELINK

[@Nothing-ui7pj]

To check wether they are fundamental we take the wronskien at the singular point right?

[@ayadalhashimy4987]

Q1. Solve the following differential equation using power series at x=0. Find the first few terms. y" + x²y + xy = 0

[@dwilmer7]

patrick didnt you say you did a more complicated example?? cant find it? thanks

[@humphrec73]

a more complex example? Is it available yet?

[@tejumadeolomola2627]

I hope you living the life you want!. Cos you are a gem

[@patrickjmt]

it is mostly good, with a few hiccups, but that is life :)

[@liquidstl]

for my Intermediate Heat Transfer Class, I have to solve T''+mT^4=0 where M = 2epsillon*sigma/k = constant..... I am in the middle of an attempted solution using power series but I don't know what to do because I cant consolidate terms since one summation has X^k while another has X^4k. Where do I go from here? Are there any alternative solution methods, etc? Thanks!!!! -Chris Downs

[@noodlefart1029384756]

i was struggling with this all year then you taught me in 10 mins lol

[@patrickjmt]

now you know where to come next time you get stuck :)

[@RosatiSamuel]

Its been awesome getting this working out here, Ive worked a lot in class and not understanding! kudos tutor, but how did you factor out the second step of the summation, little bit confused over there

[@Gobemoucheful]

So when you took the derivative of your first seires why did the summation start at n=1 instead of n=0?

[@ummarfarooq4160]

Awesome one thanks

[@stanleyho2499]

This video helped me a lot. Thank you!

[@patrickjmt]

and the cam was a $100 webcam by logitech

[@patrickjmt]

@dwilmer7 i had every intent to when i made this, but i never did yet

[@patrickjmt]

nope! c_n = c_0 / (n)!

[@moutzdx5255]

what if we have y'-2xy=0. can u answer me plzz ?? How you will write it as a single submission ??

[@ashwathamagaming7788]

Sir ur vdos r very helpful... Can u please tell me if the method u showed in this vdo is known as Frobenius method??????

[@wlthr]

no, that would be y_p=sum(a_n*x^(n+r)) with a0 != 0

[@hashtagjules631]

you should do more video on power serious solutions and frobenius' theorem please :)

[@jaybee9631]

Hi Patrick, how would you solve if the equation was non-homogeneous?

[@Blacksnowman]

Can anyone tell me how I am suppose to know when to use power series to solve a differential equation?

[@kiddcamel]

I wish I could thank you in person

[@skipunkrich]

have you posted the more difficult one yet?

[@sravanthysharma5032]

Thanks a ton patrick ur videos are really helpful

[@dwilmer7]

thank you.. one more question.. after you take derivative of first summation (n=0 to n=infinity).. why does it then change to n=1 to n=infinity?? thank you

[@rippen526]

Wow your comment was 11 years ago, 😨 .

[@RedBar3D]

How do we know that y = Ce^ax is the only solution to differential equations like these, where y' = ay? Is there any way to verify that there are no other solutions?

[@socratiat_0]

I think that there are other ways .. but the problem is I do not speak English well

[@RedBar3D]

Socrat AL-Baghdadi you could provide me with a link

[@ostensiblyquerulous]

Simon Nylund The goal is to just find solutions that span the necessary solution set space. So it is very likely a different answer does exist, but that's irrelevant, because we don't have methods for solving for them.

[@23youwatch]

is there a second example video?

[@Prashanth-yn9zd]

Is C0.e^x the only solution for that equation?

[@corbinsherman2438]

It is not very useful if you have constant coefficients but if you have variable coefficients it is a way to find a solution to the DE

[@patrickjmt]

do not be sorry! i make mistakes quite regularly!!! actually, i appreciate it VERY much when people point out my mistakes - i do not want incorrect videos! as i say, feel free to post comments or questions : ) this is how we all learn! : )

[@jonasnghifewa5541]

when n=3 we are getting C0/4! is this not supposed to be C0/(n+1)!

[@HattManDoo]

Excellent review! Thank you so much!

[@math-tutor]

wow. that video was on 2009 :)

[@roblecrystaljade]

Sir, what book do you use as reference??

[@patrickjmt]

@MizuPsi i only know how to do simple examples

[@PTNLemay]

If Co acts like an initial condition, how would we have solved it if y(0)=0? In that case the sum would have had a zero multiplying the rest of it. Which feels like it would stopped everything in it's tracks.

[@thephlebotomist]

could we see some of important ODEs that need this approach like a bessel or airy function?

[@liquidstl]

Hey could you do a more complecated example of this? thanks