Introduction to Linear Differential Equations and Integrating Factors (Differential Equations 15)

[@MorbusCQ]

After watching this video I was able to go to the gym and added some arbitrary constant to my bench press record.

[@billystewart6965]

Good one

[@billystewart6965]

ItS called a constant of integration

[@Ben-mt7yj]

Aye go blue! Also go Math 216

[@lanceward7048]

Lmfao

[@lanceward7048]

Arbitrary constant is a term Leonard uses very often!

[@marlonguerrero5680]

I am a bodybuilder and I went straight into integration.

[@quahntasy]

I am a physics grad student and I went straight to gym after watching your video.

[@pockydust3404]

You're a physics grad student and you're learning this? Isn't that a bit too easy for your level now?

[@izzy31337]

@@pockydust3404 stop being a hater, he could be recapping.....

[@Jannikheu]

@@izzy31337 or he was just searching for a motivation to go to the gym

[@pockydust3404]

@1 No not really

[@banquo4223]

@Zivojin Limanic I dont get it

[@afrolichesmain777]

Im a Mathematics major and highkey after watching your videos I started going to the gym

[@godiswatching_895]

lmao that took an odd turn

[@duckymomo7935]

I want to do that too

[@JasonWard79]

I am a gym major and these videos made me start going to the math.

[@Scapegoat0442]

@@JasonWard79 lol 😂

[@basharadel8940]

God bless you Professor, you can't imagine how many students you have been helping with your great videos.

[@sethle5777]

26:07 "If you understand, raise your hand" LOL you truly teach with such an open mind of your audience, and that is what makes you so good at explaining with such clarity

[@khbye2411]

Around 5:05 to 5:15, to bother to pay attention to and explain these kinds of small details (even if it's just 10 seconds) so as to not confuse students is what distinguishes professor Leonard as a better teacher. Thank you very much for taking your time to make all these math videos!

[@avneel]

Finally after so many years I understood why we do what we do in this situation. Always memorized this procedure but first time got to know it in depth. Thank you professor Leonard, you are a rare breed of professors who know how to make tricky things simple.

[@zaka6933]

Professor Swole

[@Peter_1986]

Professor Perfect.

[@crisbarron2044]

Dude could literally crunch the numbers. X-, Y- components are for Xtremely Yacked.

[@micahsnow346]

I am the kind of person that won't remember a formula unless I understand the logic behind it. My professor kept throwing these formulas at us and it didn't make any sense until now...thank you, Professor Leonard.

[@qwerty0458]

Same honestly

[@sidehalls]

I just hate plugging in shit without understanding what's actually going on. Not satisfying

[@joserosado4667]

My Mind Is Blown!!! You are an absolute God sent! This video is totally worth the hour of watching to understand the process behind First Order Differential Equations.

[@charlesrodriguez6276]

I think an often overlooked aspect of mathematics is the importance of clear exposition. Professor Leonard not only explains things clearly the first time, he thinks of four other clear explanations to bolster his explanatory power. This really is a tremendous help and after watching this video my understanding has vastly increased. Thank you, Professor Leonard!

[@gunterrebing1556]

Professor L. is a teacher who is passionately committed to making his students really understand the difficult concept of solving linear differential equations. So he goes to great lengths to explain every step repeatedly. A lively, even thrilling performance! G.R., Bonn, Germany

[@spcverger]

This is why I come here. My instructor completely skips the "why." When you know the "why" it's much easier to grasp these concepts. Thank you! I wish you could actually see all of us "getting it."

[@tangsoopap]

I've been retired from teaching university mathematics for 18 years now. I'm going through your DE videos to see how much math I've retained as I've aged. I'm happy to say so far so good. I am impressed at your presentation here. The engineers I taught really didn't care much for proofs but like you I did strive for students to understand. You've done a real service for current students.

[@Sonofapoop02]

I will now start saying "what the function of x is going on here?!" 😂 Thank you so much professor for this excellent explanation, I was beyond lost with the way my teacher explained it! #bestprofessorever

[@godiswatching_895]

44:08 The excitement made me laugh. I really love how even though it might be so obvious to you at this point, you're still just as amused by it. Love and appreciate the videos :]

[@Abayas.]

As a person who has a difficult time following math jargon, I was only just hanging on/following his explanation for the first while. (Basically partially lost and confused.) Then, around 30:00 everything starts coming together. Just for anyone who experiences math the same way as I do. Thanks Professor!

[@GoatzAreEpic]

I have the reverse problem where if someone waves their hands instead of using math jargon or exact meanings, it doesn't mean anything to me xxDD

[@TheOnsieWithConverse]

The Legend of Math: A Link Between Equations

[@oscarobioha595]

I have been with professor leonard from the start....started from the bottom now we here.

[@RelevantDad]

Did anyone else get excited at 44:00 when he got x^3 back again? I "learned" this technique last week in my DE class, but my professor didn't explain what was actually going on, so it made no sense to me until I watched this.

[@synner5417]

Don't even know how to express my gratitude.. You sir, help me understand all the concepts that is written very poorly in my textbooks and on the slides of my Calc course. After being able to understand how and why all these rules work, I am almost intrigued to start learning Calc for fun after my exams... That's a huge impact of you on me. You are awesome.

[@MotoMechzorz]

I'm a Junior in Mechanical Engineering, and after looking over my Signals and Systems book for next semester I realized I needed to seriously brush up on diff eq this summer. I can't believe how poorly this was explained when I originally took diff eq. Wish these videos had existed back then. You are earning every cent of my patreon money :)

[@brandoncrenshaw6813]

AMAZING... I'm watching this to prime from next semester (i.e., give myself familiarity and extra time). This would have definitely been a hang up for me. Again AMAZING lecture!!!

[@mahmoudhammmad8089]

you are the best teacher i have ever seen!! thank you

[@herboize]

i just wanted to let you know that you are helping people from south america to pass their tests, i supposed it must feel pretty good , thank you for doing this, you are a great teacher

[@ericstone213]

You are a blessing to everyone who watches your videos, and I will definitely donate some money to you. Absolutely saviour in all things calculus.

[@Maya-xs9xn]

Professor Leonard is the best teacher online by far, I tip my hat. A jewel

[@Rust_go_java]

YOU ARE UNFORGETTABLE SIR, THANK YOU FOR YOUR EFFORT YOU MADE TO HELP US. TRULY WITHOUT YOU THE APPLIED MATHEMATICS UNIMAGINABLE FOR ME!!!!!!!!!!!!!!!!!!!! THANK YOU, AGAIN AND AGAIN

[@gate2024-c5b]

such a thrilling experience , i will never forget this one for rest of my life ,thank you prof...

[@VectorSpace33]

You are definitely my favorite KZbin teacher. Thank you for all that you do. God bless you and your family.

[@alreadyghosts7727]

Professor Leonard is a national treasure.

[@jennyispoop4]

OMFG to understand the ins and outs like the anatomy of the steps @44:30 what an intellectual mini orgasm that is , all teachers should be able to do this as a prerequisite to being a teacher in the field

[@mohammedsaeed7241]

beautifully explained, the way we were taught, "oh there is an "e" in there somehow, don't worry about that, this is how it's solved, just follow thro". They made math sound like song we had to memorize, but you Sir!!!!! helped us to understand it, thank you!!!

[@jcnotnot8120]

I’m a gym Mathematics and after watching your video I went straight to the Major.

[@Jana-lg6wq]

Thank you so much for actually taking the time to explain the reason behind this method, a lot of teachers just don't care.

[@ابراهيمعبدالله-ذ9د4ق]

انتي محجبة 🧕

[@charleskostrz4681]

taking differential eq now as an undergrad for software engineering, your videos are amazing ! makes things so digestible and straightforward ! keep it up !

[@kriti4698]

26:07 🙋 understood sir, I had a lot of doubts which got clarified here in this video. Thank you so much sir! You are the best maths professor.

[@loveconomics]

You should sell a shirt that says: "What the f(x)" or "What the function of x" I laughed pretty hard lol

[@tatjanagobold2810]

Was really frustrated as I did not understand the concept throughout the entire video, but 5 minutes before it ended I had a breakthrough - now I get the concept! Thank you a lot for showing this example as well :)

[@markzho]

49:30 clicked, professor Leonard thank you for not flying through this, for slow people like me, this was extremely helpful. I'm a subscriber and making sure I press like on all your videos. I appreciate your diligence in showing your every step!

[@Wizardofaloy]

I have an exam on friday about this and omg MIND BLOWN by your explanation. Still can't understand why professors on youtube are better than the ones we get in college. Just fucking hire everyone in this website, they are the ones that deserve to be there not the shitty ones we pay to get nothing in return and have to learn by ourselves

[@aidanhughes3458]

48:00 blew my mind. glad a professor finally explained why the shortcut works!

[@Conorstew]

Finally a math prof who speaks my language!

[@abdelrahmanelbatran1164]

This is one of the most beautiful videos I have ever seen, you have turned complicated ideas into very simple and logical ideas. so, Thank you very much prof. I really hope you see this comment.

[@jcnw92]

Thank you for making the world a better place with your lessons. Love your work!

[@afafemad187]

I can never thank you enough for these masterpieces you're giving out to us for free!! I still didn't take DE, I'm supposed to take it in the coming semester, but I seriously couldn't wait to study this series, so I'm doing it in the summer for fun. Thanks a bunch!

[@mango5660]

My teacher introduced this technique as though it was obvious and explained NOTHING about why it works. Thank you so much. I'd be so lost without this video.

[@joeloffredo2367]

You're an amazing teacher. You make everything seem simple and take the time to prove it and make sure we have a deep understanding of what is going on. If only my professors were the same...

[@quackqq]

Thankyou so much professor Leonard, cannot express the joy of truly understanding maths in words. I wouldn't say you're my saviour for university exams, you're much more, in fact you have made me realised how I was forgetting the beauty and logic of maths. This equation was thrown at me in class 12, which I could never ever understand why it was so, but now 1 year later I have truly understood it. A totally new perspective and things seem so much clear, a new interest has arisen, so thankful! I know I've started late for my minor exam preparation and might not score well, but as much as I will study, I promise I will understand that much truly. So happy sir! Hope you and your family is doing good, best wishes from India!

[@quackqq]

Returning to rewatch this masterpiece, before my major exams now. Amazed and reminded again!

[@HunterNapier]

It's 4am. At one point when I realized just before you said it that it would have to be e^integral f(x) I legitimately sat up and YELLED "OH!" at my screen, prompting the roommate to come running in. Thank you so much man, I'd have flunked out by now if it weren't for you and I wouldn't have made it this far without all your videos and your awesome teaching style. Still struggling a little on tests because there's a lot of moving parts and components to put into action but the theory is really sound thanks to you.

[@xavier.antony]

Dear Prof. Thank you very much for giving us a wonderful course. I really appreciate you for spending time to prepare these videos.

[@onlygaming9652]

i wish i had professor like you mr leonard he break down everything just to make sure we understand whats going he is the best professor in the world most probably...

[@coleyoutubechannel]

Your vids are literally carrying me through Diffeq. Thank you Prof Leonard -- you are the man, this video really helped to understand what is going on with linear differential equations. Books on this subject and other vids gloss over this very important concept; to think of it as creating a product of an implicit differentiation.

[@harshchauhan4436]

I don't know if anyone has said that to you prof leonard. But the patience with which you teach. The kids who aren't even good at it will definitely be a changed student after taking your course. Love the way how you pick it up from level zero and take the student from pt A to pt B step by step

[@nta699]

The buildup to 43:58 and its result was a thing of beauty! Thanks ;-)

[@insaneoblivion8462]

Thank you , thank you , THANK YOU so much for this indepth explaination. Being a junior in highschool IB HL AA Math, we never learned this level of conceptual understanding as to why the process to solve linear differentials worked. I have my final in a few days and was just giving your video a shot, my god I did not expect this. I saw the recent update video so I'm excited for new content whenever you are back, I really am grateful for this quality of content and hope you keep it when you come back.

[@shannonmower5615]

Oh wow! This is amazing! Your lectures got me through double and triple integrations in Calc 3 (I ended up with a B!). My diff. eq. professor is good at solving the problems, but I didn't know WHY any of this worked, and where my errors were. Watching this totally clarifies what is going on! Now I can see my errors. "Finding the missing component of the product rule" is genius! I hope people realize how truly brilliant Professor Leondard is. No one explains it with this much talent!

[@CarlosTorres-vr5kc]

49:14 this was the exact moment my mind was blown. Thank you Prof Leonard!

[@SnorwayFlake]

Great video as always, you are making me feel like I am sort of stumbling upon the solution to the problem and giving enough time to let everything sink in. Now I finally fully understand what I did on an exam a year ago, instead of looking up the form to solve the diff.eq. Again thanks for the great work.

[@alexcaffri9091]

Thank you! My prof explained how to do this in steps and I had no idea what was going on or where things were coming from. So far I love your teaching style even if you do repeat yourself a lot it gives us time to grok it

[@agh1750]

He motivated/introduced the derivation/logic behind the technique so well that 10 minutes in I was able to infer exactly how it works and the role of "e" in the integrating factor before he even got to that part

[@postholedigger8726]

Not simply a better professor but a better mathematician. david

[@iancoleman6352]

My prof blazed through the explanation and then gave the explicit formula for y and literally said "You are going to memorize this" The devil is in the details. Again, thank you for your in depth explanations and emphasis on critical ideas.

[@dakotamercer1679]

I pretty rarely comment on videos but man, Professor Leonard is an extremely good teacher.

[@Maths_Magic_]

He is really ramanujan for me my teacher ,mentor I give him respect from bottom of my heart ❤

[@NotLegato]

oh, thanks a lot. very helpful. my lecturer skimmed over the topic in a single example with no little explanation of the mechanics behind it.

[@AbdeenzRock]

The only channel that i let it's ads roll till the end

[@samuelolugbemi5077]

Your one of the best teachers ive ever seen ❤

[@justiceprovider9822]

we express our gratitude to you, king.

[@talhawasim5469]

the greatest teacher of all time

[@andrewwilliams2111]

I was like "man this video is an hour long, ima skim through this crap", so I skimmed it and still didn't understand. So I started at the 25:00 and watch up to 50:00 . He literally worked a problem backwards and then forward. Amazing, it's so easy.

[@stevenpan6079]

I am CCny mechancial engineering student and this professor is the god of math. I hope he comes to CCNY to teach.

[@MrVelham]

Man you are so awesome at explaining these concepts. My Math professor didn't explain it nearly as well as you did. Keep up the great work professor "swole" Leonard. Don't take the derivative of your bench press.

[@Conorstew]

Who else raised their hand @26:08? Even though Prof Leonard cant see me I still want him to know that I understand and that he is doing such a great job :)

[@Ibooo_Suna]

you should be called the MathMan, great job prof. Leonard

[@mukonank783]

You are truly a maths god sir... I got 94% on calculus1 and 100% on calculus2 on my first year... now I'm doing 2nd year and I'm aming at an another 100%... I'm not a geneous, all my success is because of you. May god bless you and continue with your work.

[@ProfessorLeonard]

Great job!!!

[@user-fk5di4me9c]

this is by far the hardest topic and concept in math that I have ever taken. even tho the vid is only one hour, I spent at least three hours trying to understand it!

[@angkundiman2492]

39:03 "Exactly" *mysterious fascinated heavy stare* hahaha I like how you evaluate these concepts like solving murder cases. You're such a cool Prof! Thank you

[@reDMR98]

This is an awesome lecture, can't tell you how much I appreciate it.

[@ericleavitt6615]

Once I got to 45:46, I understood something I could do, but never understood why. I wish the explanation was more succinct, but thank you. I think I get it.

[@livolofsson9634]

You're so goddam amazing, and your passion for the subject is somehow forcing itself upon me which gives fantastic results. Thank you!

[@johnfoe3574]

I've tried to do the theory ahead and then I've corrected my mistakes watching the video. It worked great. Thank you!

[@iZeroPvP]

I have always had trouble understanding math when it was taught by my professors. You were by far the best one I have learned from so far. I'm watching all of your other vids after this one!

[@ThePreppar]

OMG i'm speachles at ~50", Thank you Professor! I've been strugling for weeks with this!

[@BariScienceLab]

Amazing. My god, really helped me understand. Thank you so much.

[@tuckyoubch]

Thank you SO MUCH for this video!! My professor didn't explain why this works at all and I was so confused about the concept/process until watching this :D

[@jackbrolin7709]

"I'm going to tell you WHY it works" This is why Professor Leonard the best

[@adamhallberg8252]

Very good videos. I am a math major and was at first a bit intimidated by the length of your videos, but the explanations are absolutely stellar for conceptual understanding, and if you get something on the first explanation, you can just skip forward a little bit until something new happens. Thanks a lot, Professor Leonard!

[@thatsillynikola1157]

Professor! You're awesome at what you do. It's so sad there are no linear algebra, topology, or real analysis videos. RIP me next semester.

[@The-Devils-Advocate]

How did you do?

[@eugineguan2122]

Thank you professor! No matter how long is the discussion, the good thing is you explained it so clearly and makes a lot of sense to me since in the coming school yr I will be taking D.E subject.

[@sweetlemons8695]

saved my life doing a diff eq homework assignment. thanks man

[@deltax7159]

your teaching is top notch.. there is a severe lack of great maths teachers out there. lucky to be at FSU. they have some really great maths teachers, but these videos really help me fill in the gaps in my understanding so THANK YOU!

[@noel2577]

@26:08 I raised my hand

[@sebastiangalvan5465]

Thank you very much for your videos. I'm thankful that you care about students actually understanding the logic behind the math and that you care so much to share this understanding with even those that aren't your students (even though by now I feel like I am). Thank you so much!

[@onurcandoit]

You made it understood again, and again... Cycle of maths...

[@charmer1979]

Wow aafter watching last night then thinking about this all day then I re watches the video I can now solve them with understanding rather than following the e^x routine. I am into my second year degree and was atarting to doubt myself now I feel confident again yipee.

[@johnholme783]

Thanks prof, that was Crystal clear! As a physics student differential equations are very useful to me!

[@congchen6299]

Thank you, Professor Leonard, your videos help me a lot. calculus 1 to 3, I all got an A because of your videos, you explain so so well and so easy to understand. Love your videos so much. thank you very much.

[@saurabhkujur6331]

I really feel hard to understand this concept before this lecture, After watching it very careful and patiently. I understand this concept. So, thank you professor Leonard