Since Henry Cavill is done as Superman, someone cast Professor Leonard as the next Superman. Someone who actually has real-life experience saving lives.
Thanks Prof Leonard for teaching this class this semester!! Literally saving my semester right now! You've rewired my thinking for math, I've gained a whole new interest for it, and you've helped me to fully grasp and have an in depth understanding and pass Calc 3 last semester. Also, how to take far better notes and understand complex ideas in organized ways, when other prof's have not done that so much. Thanks for all you do to help students learn more, make it fun, explain the actual "why's" and application behind so many parts of these courses that are normally barely touched on, and your encouraging words as well. You also never make students feel 'dumb' and make viewers feel like part of your class and encouraged to think and try examples to learn new things for themselves. Truly help myself and friends feel that anyone can learn it if they stick to it step by step and don't over complicate things. I depend on these as much as my actual lectures at my school, so I just wanted to say thanks, since I've been watching endless hours of your videos since Fall 16 lol - David
You've very welcome!! Thanks for watching :) And yes, I think it's terrible when students are made to feel "dumb" whether by intention or not. A learning environment is never a place for condescension or arrogance. We might joke that way to have a little fun, but when it comes to the actual learning of the art of math, patience and care are the most important concepts we can have. I'm also glad you feel like "part of the class!" That's what I strive for :) I'm glad you passed Calc 3 and best of luck in Diff Eq!!
@@ProfessorLeonard This is why you are absolutely incredible, that despite your mastery and incredible wisdom in Math, you are still humble. Thank you so much for your classes, I hope someday to be able to reward you for all you've done for me.
This channel is great, differential equations has been so annoying to learn online, my professors don’t have the same level of English fluency and struggle to convey complex concepts in clear intuitive ways. Many thanks prof Leonard
I’m very thankful for these style of videos. Since school has been online only for over a year now, all my professors have just been making videos using PowerPoint or drawing with a mouse on a screen. Watching you standing in front of a board writing it all out really makes it feel closer to being in the classroom.
Just starting to come back to maths after many years and studying at college. These videos have been fantastic at getting me up to speed and have to say better than some of my tutors at college. Thanks
I've just got an A in Cal3. I know "C" is a constant, but I had never known that C makes "parallel curves." Thank you, Professor. You are the best of the best!
I'm taking differential equations currently and your videos are incredibly helpful! I went from being confused about these concepts to having a solid grasp of how it works. I'm not usually one that enjoys math, but I have to say that I'm excited for the next videos in the series to come out!
Hello, Professor Leonard. I really appreciate all the work you have done to put videos for all different levels of math, including the high-school part as well. Since some of the people actually want to jump from intermediate to calculus level of math, they miss the final piece of the puzzle, which is trigonometry. It would be really amazing if you started series on trigonometry from beginning to end, so people can actually have no gaps between their high-school math and calculus level mathematics. I, myself have actually watched most of your intermediate algebra and statistics, to get a quick refresher, and now I am watching the Calculus videos, climbing up the ladder to this section, which I find wonderful, even though it is still out of my range. Keep up the good work, and please, think about doing trigonometry - it would be so much helpful, as there are no videos in youtube, that are explaining it nearly as good or in-detail as you do.
Thanks Mr I gat a better understanding of all calcus courses by watching ur all video rather than learning in the normal class room after I joined in one of the university of Ethiopia and makes me love mathematics more..thanks again
My professor makes me feel dumb when he teaches me exactly the same thing. I was freaking out about dropping my class, but you really saved me. I appreciate you taking the time to teach us instead of defeating us. ❤❤❤❤
My teacher yell very little on it because he made me understand the concept and it was You Yes I learnt Calculus by watching your Calc 1 and Calc 2 videos thanks for uploading them
As soon as I became a mechanical engineer, I want to meet this guy and have a drink with him. Salute to people like this. You're one of the building blocks of my success. Thank you sir.
It’s been almost 2 years since I took diff eq, I almost forgot how to do a u-substitution lmao. Thanks professor! It’s good coming back to these type of videos without worrying about the stress that comes with exams.
Professor Leonard, I know the chances of you reading my comment are low but I really applaud you for covering even trig substitution, I never really understood trig substitution in calc 2 but watching you do it in this video made it click for me in a way that I had not gotten before. Thank you so much for taking your time and explaining even what some may consider as "basic"
Professor Leonard thank you for a fantastic explanation on Solving Basic Differential Equations with Integration. Techniques of Integration from Calculus II is very important for this and future topics in Ordinary Differential Equations.
Hey Professor Leonard, I just wanted to thank you from the bottom of my heart for all the videos you have done for both calculus 1 and 2. Because of your clear and concise lectures, I was able to earn an A and a B in both calculus one and two respectively. Thank you for being the best math professor I've ever had in University. I'm definitely going to recommend your channel to everyone I know!
You should know: I am from Norway and you teach better than my norwegian teachers I've had. I wished I had you instead😁 keep up the good work! 💪💪 Math is cool
You are using your own free time to teach others. Some professors take the money and do not explain ( intentionally) and make it seem harder. They end up messing our lives and take our money without doing the job. What a shame they!
Before we start, with a dy/dx on the left of =, and some f(x) on the right, we can regard the LHS as F'(x), the derivative of the anti-derivataive function F. And we're done with the notation -- we're integrating the derivative of the anti-derivative, and use inverse relations of these operations.
I love how you are slowly but surely catching up to what I am currently learning in the Diff EQ class I am taking. I cant wait for you to get to Exact Equations, because those don't make a single bit of sense to me.
Me: *innocently watching this video because I need to learn differential equations* *My bf walks by*: Wow, try not to fall in love with Professor Math Muscles over there. >_
When attempting the first problem I foiled the """ (x - 2)^2 """ term before I took the integral. It gives me a very similar solution, but not the same solution. After integrating my answer is: """ (x^3 /3) - 2x^2 + 4x + c """ while his solution when expanded out gives: """ (x^3 /3) - 2x^2 + 4x - (8/3) + c """. I am not the most meticulous person so if someone sees my issue please feel free to point it out.
@@ProfessorLeonard Awesome! Thank you so very much for the response. I thought for a second that might be the case, but was not entirely sure. I really appreciate the videos. You have made learning Diff. Eq., as well as, reviewing a lot of forgotten Calculus much easier! Keep up the great work.
Professor, I’d seriously be more than happy to pay you for these courses! If you in future videos you do Linear Alg, Complex Analysis, etc., please (I’m begging you) make a website where you charge for them! I’d happily pay you $100 for an amazing course like this! You’ve provided too much free value: let us pay you!
I always understood the moving the dx part as a way to say that each dx, or change of x, is "modified" by a function to give you a certain dy, or change of y. It's the only way I can see it making some intuitive sense
Hey Prof. an update on my teaching... they want me to teach calculus for engineering! Never thought that would happen so I jumped at the chance! Thank you.
A friendly comment, I like how you cover the full course, 1) can you go with book, section by section, that will help us flowing with our class. 2) and you spend lots of time explaining, i personally keep skipping, it’s kinda bored, English is not my first language, and I still see you spending lots of time explaining. Please keep doing what you are doing it really helps, thank you so much. ☺️
He does indeed repeat himself, a bit too much for my liking, but I understand not everyone understands after a single repetition... he's probably used to teaching kids with the attention span of a fruit fly...
Question on the general solution being a family of curves plus "c"; how can we determine if there is a range( or what range) the "c" is limited to without affecting the derivative of the function. For example, why about cases where a function/curve would change after a constant reached a certain value etc.
Thanks Professor Leonard, should the first answer be y = 1/3(x-2)^3+1? I am guessing the exponent in y = 1/3(x-2)^2+1 is a slight mistake for other people watching the video. Enjoy learning from the legend! :)
I've never understood the logic behind trig substitutions. I mean, I'm able to do trig subs in my sleep, but I never understood how or why that's a valid piece of reasoning.
@@onemanenclave it's identical? Or equal? You substitute the trig function in replacement to its equivalent radical expression which is harder to integrate. i don't understand much either but that's my hunch haha
for integration by parts i made something i call a di chart. like this /(x^2*e^x) derivative integrate D | I +x^2| e^x -2x | e^x +2 | ex youll never forget it
Regarding your opening. When x is the independent variable and y is the dependent variable then f(x) = y. So your dy/dx = F(x) translates to dy/dx = y. IOW, there is both a derivative of a function and the function itself in a differential equation, right? dy/dx = c (constant) isn't a differential equation, is it?
I came for the tutorial but I stayed for tutorial+Professor Leonard.. does this man have any idea how beautiful he is ;) well I better go back to my research, I have seen enough beauty for today
![Solve & Verify Differential Equations by Integration - [2]](/img/n.gif)
Since Henry Cavill is done as Superman, someone cast Professor Leonard as the next Superman. Someone who actually has real-life experience saving lives.
This wins commenting.
Is it a plane? Or is it a surface? No, it's Professor Leonard! (cue Superman theme here) :)
I love this.
@@bonbonpony is it a plane? is it a surface? no its riemann space
[35:40]. Professor pretending he doesn't know what arctan(0) is to try and make us noobs feel better. What a legend 👍
I love him for that. It does not matter what he knows. The important thing is he is not making us feel stupid, and I'm all here for it
Thanks Prof Leonard for teaching this class this semester!! Literally saving my semester right now! You've rewired my thinking for math, I've gained a whole new interest for it, and you've helped me to fully grasp and have an in depth understanding and pass Calc 3 last semester. Also, how to take far better notes and understand complex ideas in organized ways, when other prof's have not done that so much. Thanks for all you do to help students learn more, make it fun, explain the actual "why's" and application behind so many parts of these courses that are normally barely touched on, and your encouraging words as well. You also never make students feel 'dumb' and make viewers feel like part of your class and encouraged to think and try examples to learn new things for themselves. Truly help myself and friends feel that anyone can learn it if they stick to it step by step and don't over complicate things. I depend on these as much as my actual lectures at my school, so I just wanted to say thanks, since I've been watching endless hours of your videos since Fall 16 lol
- David
You've very welcome!! Thanks for watching :) And yes, I think it's terrible when students are made to feel "dumb" whether by intention or not. A learning environment is never a place for condescension or arrogance. We might joke that way to have a little fun, but when it comes to the actual learning of the art of math, patience and care are the most important concepts we can have. I'm also glad you feel like "part of the class!" That's what I strive for :) I'm glad you passed Calc 3 and best of luck in Diff Eq!!
Professor Leonard Thank you!!
@@ProfessorLeonard This is why you are absolutely incredible, that despite your mastery and incredible wisdom in Math, you are still humble. Thank you so much for your classes, I hope someday to be able to reward you for all you've done for me.
This channel is great, differential equations has been so annoying to learn online, my professors don’t have the same level of English fluency and struggle to convey complex concepts in clear intuitive ways. Many thanks prof Leonard
Your language and method of teaching is really appreciable
I’m very thankful for these style of videos. Since school has been online only for over a year now, all my professors have just been making videos using PowerPoint or drawing with a mouse on a screen. Watching you standing in front of a board writing it all out really makes it feel closer to being in the classroom.
in the first 2 minutes he's explained what differential equations actually are, better than I learned in all of engineering school
Just starting to come back to maths after many years and studying at college. These videos have been fantastic at getting me up to speed and have to say better than some of my tutors at college. Thanks
I've just got an A in Cal3. I know "C" is a constant, but I had never known that C makes "parallel curves." Thank you, Professor. You are the best of the best!
I'm taking differential equations currently and your videos are incredibly helpful! I went from being confused about these concepts to having a solid grasp of how it works. I'm not usually one that enjoys math, but I have to say that I'm excited for the next videos in the series to come out!
Hello, Professor Leonard. I really appreciate all the work you have done to put videos for all different levels of math, including the high-school part as well. Since some of the people actually want to jump from intermediate to calculus level of math, they miss the final piece of the puzzle, which is trigonometry. It would be really amazing if you started series on trigonometry from beginning to end, so people can actually have no gaps between their high-school math and calculus level mathematics. I, myself have actually watched most of your intermediate algebra and statistics, to get a quick refresher, and now I am watching the Calculus videos, climbing up the ladder to this section, which I find wonderful, even though it is still out of my range. Keep up the good work, and please, think about doing trigonometry - it would be so much helpful, as there are no videos in youtube, that are explaining it nearly as good or in-detail as you do.
Thanks Mr I gat a better understanding of all calcus courses by watching ur all video rather than learning in the normal class room after I joined in one of the university of Ethiopia and makes me love mathematics more..thanks again
Hi sir
Great job ....my words won't do justice....for your valuable teaching ...
All the best ...
Best wishes from India..
I love the fact that leonard reinforces the underlying concepts just enough so it isnt like review but enough to understand the next step.
My professor makes me feel dumb when he teaches me exactly the same thing. I was freaking out about dropping my class, but you really saved me. I appreciate you taking the time to teach us instead of defeating us. ❤❤❤❤
Is linear algebra on the horizon? Thank you so much for these videos!
My teacher yell very little on it because he made me understand the concept and it was You Yes I learnt Calculus by watching your Calc 1 and Calc 2 videos thanks for uploading them
As soon as I became a mechanical engineer, I want to meet this guy and have a drink with him. Salute to people like this. You're one of the building blocks of my success. Thank you sir.
Same bro
It’s been almost 2 years since I took diff eq, I almost forgot how to do a u-substitution lmao. Thanks professor! It’s good coming back to these type of videos without worrying about the stress that comes with exams.
Professor Leonard, I know the chances of you reading my comment are low but I really applaud you for covering even trig substitution, I never really understood trig substitution in calc 2 but watching you do it in this video made it click for me in a way that I had not gotten before. Thank you so much for taking your time and explaining even what some may consider as "basic"
Sir Leonard ..you are the Superman for me ...superman saves people from falling from the building..you saved me from not understanding maths ..
Professor Leonard thank you for a fantastic explanation on Solving Basic Differential Equations with Integration. Techniques of Integration from Calculus II is very important for this and future topics in Ordinary Differential Equations.
Your videos and explanations are very long BUT you explain things really well and in depth. So it's worth it if you have the time.
Virtual training is the way to go. Thanks Leonard.
I breezed through Calc 2 because of your videos, so I decided to tackle Differential Equations over the winter! Thanks for the amazing videos!
Loving this channel, this professor and differential equations
I miss the audience. It's very amusing whenever you ask a question for the class and then. . . silence.
I love this comment so utterly much
@@Peter_1986 Not really angry. He just wanted to involve them to make sure they were following along.
Hey Professor Leonard,
I just wanted to thank you from the bottom of my heart for all the videos you have done for both calculus 1 and 2. Because of your clear and concise lectures, I was able to earn an A and a B in both calculus one and two respectively. Thank you for being the best math professor I've ever had in University. I'm definitely going to recommend your channel to everyone I know!
You should know: I am from Norway and you teach better than my norwegian teachers I've had. I wished I had you instead😁 keep up the good work! 💪💪 Math is cool
Yes I understand, it’s an equivalence of operators. It’s in my A level maths textbook.
Can we just take a moment to appreciate just how gorgeous this man actually is.
You are using your own free time to teach others. Some professors take the money and do not explain ( intentionally) and make it seem harder. They end up messing our lives and take our money without doing the job. What a shame they!
Before we start, with a dy/dx on the left of =, and some f(x) on the right, we can regard the LHS as F'(x), the derivative of the anti-derivataive function F. And we're done with the notation -- we're integrating the derivative of the anti-derivative, and use inverse relations of these operations.
I love how you are slowly but surely catching up to what I am currently learning in the Diff EQ class I am taking. I cant wait for you to get to Exact Equations, because those don't make a single bit of sense to me.
Me: *innocently watching this video because I need to learn differential equations*
*My bf walks by*: Wow, try not to fall in love with Professor Math Muscles over there.
>_
my husband screams "OOOO Yeah professor Leonard, teach me differential equationsssss" as he's passing by.
@@salmaalzuhd hahaha even they can't deny it
@@salmaalzuhd I hope your husband says that with a Randy Savage macho man voice OOO YEAH BROTHER
This Prof saved my life, thank you!!
my professor was very bad this semester so you literally saved me, thank you very much
Thanks for this, I’m in cal 1 and am fascinated with now what I can dig into. Your explanations and teaching is great
When attempting the first problem I foiled the """ (x - 2)^2 """ term before I took the integral. It gives me a very similar solution, but not the same solution. After integrating my answer is: """ (x^3 /3) - 2x^2 + 4x + c """ while his solution when expanded out gives: """ (x^3 /3) - 2x^2 + 4x - (8/3) + c """. I am not the most meticulous person so if someone sees my issue please feel free to point it out.
If I'm reading this correct, you can combine the constant -(8/3) with the +c to obtain your +c. Same answer :)
@@ProfessorLeonard Awesome! Thank you so very much for the response. I thought for a second that might be the case, but was not entirely sure. I really appreciate the videos. You have made learning Diff. Eq., as well as, reviewing a lot of forgotten Calculus much easier! Keep up the great work.
Watching this @3:30 AM and tomorrow isn't even my exam...He is too good ya
That trig in the 4th example was awesome!!
Thank you for all you do Professor Leonard.
Thank you Professor Leonard
Professor, I’d seriously be more than happy to pay you for these courses! If you in future videos you do Linear Alg, Complex Analysis, etc., please (I’m begging you) make a website where you charge for them! I’d happily pay you $100 for an amazing course like this! You’ve provided too much free value: let us pay you!
He has a patreon if you want to support him, it is in the description
I always understood the moving the dx part as a way to say that each dx, or change of x, is "modified" by a function to give you a certain dy, or change of y. It's the only way I can see it making some intuitive sense
Hey Prof. an update on my teaching... they want me to teach calculus for engineering! Never thought that would happen so I jumped at the chance! Thank you.
Fantastic!!!! I love teaching calculus. Have fun :)
You are the best, thank you so much for doing these videos they are a huge help.
This series will be the saver of my differential equation course this semester! Thank you very much, Professor Leonard!
The earlier big videos were even better
This also good
Professor Leonard is the best!
Good U substitution. The X.dx converted to du/2
Excellent work Prof, I share your videos with my students.
thank you for explaining
This man is unbelievable
Tried my ass off to memorize or get the stuff about trigonometric integrals. Thanks, your explanation really made me get it once and for all
Thank you Professor Leonard!!!
how could one person have it all?
Looks, smarts and is very well spoken???
Watching these has increased my gains
God be pleased with you,İnşAllah,Amen🤲
A friendly comment, I like how you cover the full course, 1) can you go with book, section by section, that will help us flowing with our class.
2) and you spend lots of time explaining, i personally keep skipping, it’s kinda bored, English is not my first language, and I still see you spending lots of time explaining. Please keep doing what you are doing it really helps, thank you so much. ☺️
He does indeed repeat himself, a bit too much for my liking, but I understand not everyone understands after a single repetition... he's probably used to teaching kids with the attention span of a fruit fly...
It is really amazing the way you teach! you are awesome!
"If you don't have the plus c how are you gonna find it? It doesn't make sense" :,D I love you
Wow. This is the first video of yours I watched from start to end. Gonna check out the trig sub for more practice.
"I don't like suspense in Math." If only more teachers thought this way!
Professor you have a good method for the study mathematics
thank you for these videos! youre saving me right now
he is
Math GOD !!!!
Thank you professor Leonard. I tried reading textbook and slept off.
Question on the general solution being a family of curves plus "c"; how can we determine if there is a range( or what range) the "c" is limited to without affecting the derivative of the function. For example, why about cases where a function/curve would change after a constant reached a certain value etc.
Thank you professor
Thanks Professor Leonard, should the first answer be y = 1/3(x-2)^3+1? I am guessing the exponent in y = 1/3(x-2)^2+1 is a slight mistake for other people watching the video. Enjoy learning from the legend! :)
Always brilliant explanation
Is this kind of problems also called as variable seperable questions ? By the way thanks for your labour and contributes to our academic life.
Its crazy how he pointed out the EXACT easy mistake I made on the last problem and said be careful not to make little mistakes on the easy stuff...lol
ever watch a video and go wow. (this is me as a phd students, with a first class masters in aerospace engineering)
where are your posted numbers? mine aren’t very high but my bro split is going🔥🔥🔥
Enthusiastic about the rest of the topics on diff eq. By the way, why C(constant) is negative on the second example. I got positive 125/3
I want to build statue of your balls sir. You are a life saver thank you very much.
thanks, superman
Sir you are amazing
Keep going
love your videos. Think you'll ever cover linear algebra?
Why the hell am I watching this? I don't even take calculus
lmao
thank you Brofessor!
I think professor Leonard can try Live video for every lecture to communicate with students.
I've never understood the logic behind trig substitutions. I mean, I'm able to do trig subs in my sleep, but I never understood how or why that's a valid piece of reasoning.
Fled From Nowhere I think it's just to make the integration process more efficient/possible. I mean, that's what u-subs are for.
Geo Polos That's the goal, but that doesn't explain why it's logically valid.
@@onemanenclave it's identical? Or equal? You substitute the trig function in replacement to its equivalent radical expression which is harder to integrate. i don't understand much either but that's my hunch haha
When u teach something it feels like i am taking math classes from superman of Jupiter 😁😁😁
Great job sir
24:04
square root of something squared is equal to it's absolute value
this man is my god
for integration by parts i made something i call a di chart. like this
/(x^2*e^x) derivative integrate
D | I
+x^2| e^x
-2x | e^x
+2 | ex
youll never forget it
very helpful, man
Sir thank you!
Regarding your opening. When x is the independent variable and y is the dependent variable then f(x) = y. So your dy/dx = F(x) translates to dy/dx = y. IOW, there is both a derivative of a function and the function itself in a differential equation, right? dy/dx = c (constant) isn't a differential equation, is it?
Right because if a rate is constant, is it truly a rate? ... It is not changing over time...
This lecture so hard, I couldn't understand it
6:26 but doesn't the left side when you integrated "1dy" also have its own +C?
magical sir!
Why don't you add the constant to the antiderivative of dy?
You're the BEST..
You are a Legend 😭🙏
What book are you using for the lectures on Differential Equations?
I came for the tutorial but I stayed for tutorial+Professor Leonard.. does this man have any idea how beautiful he is ;) well I better go back to my research, I have seen enough beauty for today