Solving Linear Differential Equations with an Integrating Factor (Differential Equations 16)

[@mariambobat]

The teacher who made mathematics a thing of beauty for the ones who were close to giving up!

[@AlphaNiner-yk8kh]

Says he’s not gonna reteach the concept *does it anyway* I love professors like this

[@88GTA]

Was about to comment this same exact thing lmfaooo

[@TheDemonnat]

Same

[@ProfessorLeonard]

Remember that when we have to divide by X, we should be stating a condition we are placing on the differential equation. TYPICALLY, this is x>0. I will take this for granted in this video. It SHOULD be stated when rewriting a differential equation. So, even though I take it for granted (and probably should have written that), YOU SHOULD DEFINITELY be writing that condition on your paper. This condition, x>0, ALSO lets us do things like take the integral of 1/x and not have to worry about absolute value.

[@Dan-bo4pc]

Could you possible do a video on how to solve differential equations using a graphing calculator

[@learning.inquiry]

Ahhh i was having headaches thinking on why you were solving any ∫(a/x)dx as ln(x^a) instead of ln|x^a|. I´m confused because we are learning this topic in "Calculus 2 for Economics" and they taught us a different method for linear diff. equ. which i find far more complicated. I´m wondering if my teacher would consider it wrong to place x>0 as a restriction for the kind of excercises that we work out; it is clear that dividing by x implies x≠0, but x>0 is a stronger restriction, though in economics usual variables such as price and quantity are always non-negative (anyway, i´ll ask him). What do you think Leonard?? Is there a way that i could mess up my solution by placing that kind of restriction? By the way, great work!! Cheers from Sudamerica.

[@frankvalen2147]

I Dont understand what you meant by that when we are dividing by x that it is x>0 this. Why is that. Is there any of your video where you mentioned this that i could watch to better understand what you meant.

[@lorenaegusquiza8004]

@@frankvalen2147"x" has to be greater than zero because if not dividing by zero would make the fraction undefined. So you should specify that is is greater than zero. Also, like he said, it will allow us to drop the absolute value when taking ln since x can't be negative (because we are stating x>0) . It just makes things for us.

[@VMGAMEPLAYS]

1:10:19 Might be the highlight of Professor Leonard. He told us, I'll leave you to it to try it at home and then just couldn't resist but to show us how to do it so he made sure we actually understood and got the right answer. He cares so much about his students, he is a what every professor should be like!

[@iayoub777]

After so much anger on deferential equations. Professor Leonard made me laugh at 1:10:20

[@isobar5857]

This man is exceptional. There are those who know what they ' teach ' and then there those who can 'teach' what they 'know' ...and there is a world of difference between the two. Professor Leonard is the perfect example of the latter. He is heaven sent...may he live long and prosper.

[@chukwuemeka_2005]

Professor Leonard changed my life ever since i met his youtube channel. Maths topics that I found very abstract is now easy to my understanding because of you. When i finally become a prominent person in life, i would make sure to reward YOU in a priceless fashion. I owe you big time

[@Sonofapoop02]

I knew he couldn't resist solving the last problem 😂 you're the best professor Leonard!

[@ricocollado1496]

42:12 Professor be spitting bars 😮🔥 "sine of pi is 0 e to the 0 is 1 1 minus 2 is negative 1 C equals negative 1"

[@huligangaeddhaerika-nonan9963]

when I want to look for certain math tutorials in youtube, I search your channel first to see if you have a video about it. I wish all teachers are as passionate as you

[@davidjhyatt]

I can't understand my notes, I watch video, notes now clear. Thanks!

[@cheez1903]

You must get this a lot, but you are the best person to teach this on youtube, all other videos don't go over everything as concisely as you do. Merci

[@lastonehabombe7210]

You've unlocked all struggles I had professor, you're really great

[@Aymen0627]

For the example starting at 54:17 , I paused the video and immediately did it by separable equations technique. Just happy that I caught that and still got the right answer!

[@lindseytreweek8771]

I don't always understand it all at first, but seeing the gaps fill in as I keep practicing is so rewarding! Thank you Professor Leonard!

[@dougfarrell5091]

Prof Leonard, first off I want to express my appreciation for you taking both the time and effort out of your, I’m sure busy schedule, to create and post these great videos. It is obvious that you enjoy teaching and want to make sure your students do absolutely understand what you are teaching. I spend quite a bit of time on KZbin taking courses and I find your teaching methods very detailed, organized and easy to follow. I had many math classes while working on a physics and math degree about 50 years ago and realize that I have forgotten how to handle many problems. I enjoy going back and reviewing subjects that I did not get to cover previously or something that interest me that I want to understand better. I decided a few months ago to go back and take some more math & engineering courses. While following some MIT lectures on electromagnetism the lecturer wrote a 2nd order differential equation to solve on the blackboard. I quickly realized that I did not remember anything about how to find the solution to this type of problem. Consequently I decided to review how to solve these types of problems so that I will be able to better understand the math and science of the physics course that I was taking which led me to this course of yours. I also plan to retake your calculus courses to get myself back up on this subject again. So that I am understanding this concept of integration factor, one question I have is on the example problem: DY/DX + 3Y = 2X e^ -3x, when you integrate [e^3x] on the left side, wouldn’t that be equal to [e^3x/3] rather than just e^3x? In this method I thought we were finding the integral of P(Q) which I thought was [e^3x] and not the quantity [3e^3X]. What am I doing wrong in this problem?

[@loopingmoon1444]

Hey Doug, I might not be able to explain it as well as he could but I'll give it a shot either way. He's not actually integrating E^3x. When he writes Dx[e^3x*y] that means its the derivative of the function in brackets (which is calculated with the product rule mentioned in the video). So when integrating he integrates the derivative, making it the function itself. I hope this helps (and also hope im correct as im just a student of calculus and still learning)! Oh wow, this is 3 years ago so it really isn't useful

[@benjaminewers5410]

at 58:48 "we're going to need a U-sub". You do not actually need to go through the trouble of integrating that, because it has already been done for you. The only difference between P(x) and Q(x) is a negative, and so you can just add that negative into the already integrated form. Hope this makes sense.

[@emirhaninanc]

I am really appreciated for your videos. You are not only a teacher, but also a chance for us who doesn't have an instructor like you. Thanks a lot!!!

[@didriksoderstrom9891]

Professor, this is legit the third class in a row you are saving my grade in. Thank you.

[@josetakapele]

Resources like this make me feel like it’s going to be okay and I can pass the class.

[@saeedussalehin]

His teaching style is Amazing.

[@actualBIAS]

I'm watching your series now for a few days in a row and I'm amazed. THANK YOU SO MUCH!!!

[@krealillen8689]

Thanks for all of your time and dedication, professor! Just wanted to note that the last problem is separable as well if you subtract 'y' from both sides and then factor out y on the right-hand side.

[@Ap-ui6kk]

No ego, only love of teaching. I love this man ♥️

[@paulchavez5363]

You don’t need college if you start learning from this guy

[@spanishlanguageeducational3737]

The drills, the drills... very good. You may not get it the first time, but Professor Leonard mercilessly drills it into your head.

[@gamegirl3355]

best teacher in the world

[@wildchild1161]

So im currently in a cal 2 class and my teacher in essence stated " just memorize it and wait till diffy q for a better explanation " . I was like ... looks like its time to binge watch Professor Leonard and get to the bottom of it :D Thanks for taking the time to make these videos and going the extra mile to work through And explain example problems. Im not sure if all colleges make math/science majors take " mathematica " but the one i go to does . The teachers never seem to explain the syntax of mathmatica and I was wondering if you could make a few tutorials on using mathematica .

[@BillyPJudd]

I knew you would have to finish that last one - AWESOME! I needed a laugh. Thank you :)

[@frankie5928]

My god, I just want him to wrap those massive arms around me and tell me that everything will be ok

[@Chapz725]

@Zzzzeyd fromkie might be a girl bro or maybe he is actually gay.

[@wawaT100]

lmao..ahahahahahha

[@VndNvwYvvSvv]

Why do you people have to force that into everything? That's what people are disgusted by.

[@RaeTheActuary]

thanks so much man for everything, was struggling to understand and now am really good at it

[@kyleblankenship1682]

Words cannot express how happy I am that I found this channel.. Professor Leonard is the best! Thank you so much! 😁

[@tgl2343]

It's like watching Bob Ross paint. The only difference being it's Mr. Leonard instead of Mr. Ross, and he's solving differential equations instead of painting. Every teacher and professor should have this kind of interest and enthusiasm when teaching.

[@trseagro]

And a happy little 'C' goes here....

[@logan6576]

At 53:00 the equation actually is separable. Just factor into (1+x)(1+y)

[@woodsmith_1]

Dude, you're awesome. Thank you so much for your work in making these videos. You've changed a lot of lives for the better. Hope everything is going ok in your world!

[@dm_drums7646]

literally the only professor that has been able to teach me!!!🤝🤝

[@greengirl4985]

this man will be in my heart forever I love you

[@alishad9777]

Great videos! Thank you Professor Leonard for all you do and for explaining the why behind the how. @ 1:01:30 I believe the solution can simplify easily through exponential rules to y = e^(2x) - 1

[@math-toss]

Thank you , you make mathematics more joyful . our almighty be with you all the time.

[@alexg91024]

This video cures cancer. Just subbed on patreon

[@samimahassan1716]

the ending was soo cute you won my heart once again

[@volkanulker6432]

we can use first tecnique for question at 55:17. First techinque makes steps shorter

[@hannahgibbs7523]

I literally was thinking about dropping my diffy q class before watching this thank u professor leonard

[@iovi558]

When I saw that 9 I thought I was about to be learning imaginary numbers... that had me sweatin

[@amk1108]

my guy drew a lolipop

[@PedroOliveira-ez2ni]

I wanted to do the exercise beforhand but I didn´t know if that was a number or not hahahahha

[@sommagar1259]

@@PedroOliveira-ez2ni Yap same here hahaha I waited till he said it.

[@abaundwal]

You show us a lot of examples. I love that! Keep up the good work professor. :)

[@ahuerdogan9954]

Next time I go to the gym, I first pay attention to my "form"

[@joshuageorge7997]

thats gonna be a very "integral" part of the process

[@dkayal227]

I must echo the other positive comments. I definitely would not have been able to finish my homework successfully with what the University teaches me. Thank you for making these awesome videos!!!! I will certainly contribute through Patreon :)

[@N7_CommanderShepard]

It’s nice to refresh on this material having took the course two years ago. Makes me glad to know that I don’t have to study for this anymore lmao.

[@retiredhater]

i have been watching your videos since calculus 2, thank you so much!!

[@hmwavy9794]

I genuinely want you to understand how much of a god send you are. Your the one person I would happily watch ads on youtube videos if you had them. You fuckin LEGEND. G.O.A.T.

[@Razzer_racer]

29:35 I thought he said "Thats an obsidian i". Everything I thought I knew about math went out the window there for a few seconds.

[@nikolozperadze4887]

Amazing Stuff! Great work connecting theory with concrete examples.

[@mejaiss392]

This guy is a legend. Im actually understanding

[@itamar.j.rachailovich]

One of the best professors I’ve ever heard!!! I wish in Israel we had professors like him

[@bobduncanduncan2742]

You are a gift from God. Thank you.

[@kennethq.mavuso3993]

I love you man...

[@danielurdiales2856]

The ending was the best part!

[@Alex-zm6xi]

thank you for the lecture mr clark kent

[@vincentrusso6120]

Thank you for videos currently in Multivariable now and have gotten all A’s from watching your lectures.

[@caesaryumury5839]

for god sake Leonard if you could see what my book makes me go through to solve this.... I am crying of happiness

[@mfarro23]

For the example problem at minute 25:39, how did he go from 10 sq rt x/x to 5/sq rt x? Can anyone let me know? thanks

[@mateocamargo7437]

Omg my birthday is October 24th!!! Your videos are very helpful :D

[@lxgolden1880]

This man is my hero 🦸‍♂️

[@dpmike32819]

Thank You for taking the time and explaining it thoroughly

[@CedarAce1000]

At 55:00, why not factor by grouping all the way and make it separable as y'=(y+1)(x+1)?

[@aaronguerrero592]

such a good guy 🤟🏽🤟🏽 51:10 , not necessary at all

[@MaxKraydich]

THANK YOU PROF YOU ARE THE GUY

[@alexatg1820]

I don't know why but I'm so into differential equations

[@mejaiss392]

could someone explain at 30:58 what happend to that ix^3 ? was that i looking thing supposed to be 9 or am i missing smt

[@BillyPJudd]

I could leave a really long comment here, but I will just say thank you - oh my glob, thank you.

[@ritesha8050]

Man you are a life saver

[@seaque.]

Professor Leonard at the end: “I am sorry, I couldn't resist.” -Dr.Schultz from Django Unchained

[@markwiley3151]

Clark Kent still being Superman while he’s Clark Kent.

[@Spongepop1357]

Ahahaha I laughed at the last part where he popped up with the answer saying, "just kidding!" Thank you for that passion, I was able to check my answer to the last question and see what I was missing.

[@Padthai_Shrimp]

Thank you for working it out on the last problem!

[@alessandronova7761]

at 29:30 I got so scared because of the messed up 9 I thought it was a mathematical stick figure that had its own laws and properties that I would have to learn in order to do the problem

[@markkennedy9767]

At 31:45 why isn't it ln |x| in the integral. This would then give a plus and minus in the solution for y wouldn't it

[@escuetajustinecarolines.5337]

Thank you so much Professor Leonard!

[@nicksardelli890]

at 51:05 the integral of sinxcosx is -1/2cos^2(x) not 1/2sin*2(x). Unless those are the same thing??

[@gevodem]

Hi Professor Leonard, at the 25:07 mark, shouldn't the right side of the equation be 5*sqft(x)/x instead of 5/sprt(x)?

[@loveconomics]

Professor Leonard I have a question about the last problem. When you are checking your work-- Our result is xe^(-3x)y=C; Where does the 1/x come back? The step prior to that was e^(lnx) which left us just x. But when we check our work derivative of x gives us 1. I'm a tad confused on this last step. Thanks for all your help! Time stamp: (1:10:23)

[@hey12delila]

Thank you very, very, very much.

[@d4mby749]

Insane video, I learned a lot new stuff.

[@ayeshaali3990]

tbh havent seen someone like him before

[@dharmanshah1239]

Salute to your dedication sir!!

[@Lauren___]

Thank you for this!

[@alejandroperera9924]

at 36:00 , shouldn't the right hand side be (9/2)x^(-1/2) ????

[@tristenhowe2886]

14:37 So what hes saying is that the stuff in the brackets is the stuff thats not touched by derivitives?

[@f-22raptor25]

the stuff in the bracket, you can get by the product rule, but you dont need to just do p(x)y

[@N7TWL]

Flag on the play at 25:10. I do not think it should be improper to leave a radical in a denominator. Yes, I know - convention says to always rationalize the denominator, or at least that is what we are taught. However, there is no compelling mathematical reason for doing so. The convention, as near as anyone can tell, started some 350 years ago when someone was creating a reference table, and in order to use the table, there could not be a radical in the denominator, and we have been blindly following that convention ever since, even though the table hasn't existed for more than 300 years! Your example problem illustrates a very good reason for not rationalizing the denominator. Since you teach integral calculus, you know that many times we need a radical in the denominator of a U-sub to cancel a radical in the numerator of the integrand. And how about a case from trig, where we are solving right-triangles and somebody writes that the sine = 2*sqrt(2) over 2. The person is likely to determine from that information the the side opposite the angle is of length 2*sqrt(2) and that the hypotenuse is of length 2. Both lengths are incorrect. The side opposite is of length 1 and the hypotenuse is of length sqrt(2), because it turns out that the triangle is a 45 degree triangle with both sides being of length 1 and a hypotenuse of length sqrt(2). I am not saying that we should never rationalize the denominator, because there are some cases where it is necessary. That's my 1.414 cents worth anyway. Great DE videos, Prof. Leonard. I am really enjoying them!

[@Pete1711]

Thank you, really helpful examples!

[@tariqkhan3906]

@ 43:20 isn't that a non linear function as the dependent variable y is multiplied with a transcendental function.??

[@mphobohlolo5557]

proff leonard you just the best .....

[@mausplunder5313]

after factoring the last problem can be done by seperation too

[@nafsunzakir8671]

I have been following you since my calculus 1 course, you have been the biggest help throughout calculus 1 ,2 and 3 now I am currently following your differential equation playlist and I have problems with EDE exact differential equations can you please make a video on that Professor. @Professor Leonard

[@ProfessorLeonard]

coming up soon!

[@nafsunzakir8671]

@@ProfessorLeonard thank you soo much Professor!! looking forward to more videos on differential equations .

[@robertbrandywine]

Whew! You kept going faster and faster!

[@JackHou-vw7hs]

48:36 why are we dropping the absolute value sign of sin x?

[@jamesbyrd405]

Because the initial condition tells us what the actual sign is that is satisfying the absolute value condition

[@dildobaggins2759]

When we cancel the e values we have absolute value x = x. the absolute value is there because of the natural log when it has disappeared there is no need for absolute value of sinx.

[@flesh2dust402]

For any future people wondering this, the answer is domain restrictions. He talks about it in the next video

[@Jimmyillusionz]

Thank you prof

[@saketshah7160]

U r quite underrated

[@dildobaggins2759]

Proffesor leonard i think dy/dx= 1+x+y+xy is se[erable you can factor by grouping surely and then put all x on one side and y's on the other and solve...?

[@accensa]

6:05 building your product

[@SachiniIroma]

at 23:49 why didnt he divide x^2 by x^5