For this presentation he's wearing a long-sleeved shirt without the sleeves rolled up. Good idea - this is serious subject matter, and we students can't be distracted by those biceps.
I am an electrical engineering major and I can legitimately say you saved my life and kept me from having to repeat my Calc 4 class. Thank you so much for the explanations!
@@abdallababikir9154 Yeah that's pretty weird. I've heard some schools mix linear algebra with ODE and call it Calc 4. Calc 4 at my school is just Vector Analysis.
I'm taking Calc3 and Diff Eq at the same time currently, and if it weren't for your videos, I know I'd be failing this year, because my professors are basically reading the textbook to us. I just wish I knew your channel existed when I took Calc1 and 2, I would have gotten better grades. I'm sharing your channel with everyone I know.
So I watched this video and the other homogeneous equation video yesterday night and this morning and was able to figure out 3 of my 4 exam problems (open notes). I want to thank you Professor Leonard for your videos because I would definitely have not been able to do my exam without your in depth explanations and tips 🙌🏽❤️
@@ProfessorLeonard your videos + Slader.com have saved me and are allowing me to pursue a career that I never thought I would be able to because of the math requirements.
I scored pretty good marks in maths in my initial semesters because of him... when he had stopped making vedios i had no choice so left maths and took another subject as minor.. my syllabus for this sem was differential equations even though I dont have the subject anymore I still watch his vedios 😁😁
Professor Leonard, you are the reason why I will pass my Calculus 3 class and my Differential Equations Class. Thank you very Much! I wish my Professors were as good as you at explaining the work, you make math fun! I Wish I could have been in your classes. Kind regards from South Africa.
Just a thought for the first problem at (6:24). If v = y^(-2), then dv/dx = -2y^(-3)dy/dx. If you divide the whole equation by y^3 you will be able to substitute in (-1/2)(dv/dx) directly. From there you can put it in linear form and solve. Saves you from all of the fraction exponents. Hard to explain in text but you essentially do two substitutions that gets you to dv/dx +2v= -2 in less steps. That's actually the brilliant thing about using substitution methods.
These computations can be simplified. Indeed, set v=y^{-2}, you get that dv/dx=-2y^{-3}dy/dx. Then divide both sides of the equation dy/dx - y =y^3 by y^3. You get y^{-3} dy/dx -y^{-2}=1. So, (-1/2)dv/dx-v=1 or dv/dx+2v=-2 or dv/dx=-2v-2. This is a simple separable equation dv/(-2v-2)=dx. Now integrate both sides: (-1/2)ln(|v+1|)=x+c. This implies that ln(|v+1|)=-2x+C. Finally, apply the exponential and solve for v to get v=-1+Ae^{-2x}.
Played all your videos at the speed of 1.5 and really enjoyed your marvel. I absorbed everything you imparted like a sponge Professor Leonard. You have helped me get a straight A in Math courses. Keep up your excellent work, Professor!
Professor Leonard - "He who is only an athlete is too crude, too vulgar, too much a savage. He who is only a scholar is too soft, too effeminate. The ideal citizen is the scholar athlete, the man of thought and the man of action." ~Plato
Even after I'm done with the diff eq class this semester, I will anxiously wait for the rest of your lecture videos. Hands down the best explanation anyone has given on Bernoulli's.
Yes..., yes you make sense!!!! The hand-waving is yet explained again!!! Thank you, Sir!!! This beats the heck out of a zoom white board where all we get is a non-rigorous proof (because coming from Calc III Students know what that is....not at all), of Bernoulli's DE's, then thrown into examples where the product rule is bypassed when we learned it, LIKE, yesterday. As a student, I do not need five shorthand examples in chicken-scratch on a digital whiteboard over zoom, I need 1 or 2 examples that dive into the details as you do in your videos. You're the Clark Kent of Mathematics, yet be the one who has found the Kryptonite of college paid professors that lack a basic understanding of how to actually teach what they understand themselves!
I'm so so thankful for this really. My prof introduced diff eqs, its properties, separation of variables, linear DEs and integrating factors, and bernoulli's equation all within a span of 40 minutes and i could barely understand anything. You're a lifesaver
Why make it simple, when you can do it the hard way...? Seriously Leonard, it took you 19 minutes (!) to state what you would obtain if you wrote dv/dx, and replaced dy/dx with the right-hand side of your the equation to be solved, instead of expressing y in terms of v (by means of a fct. that has 2 branches!)
I just wanted to mention that at 54:33, where you get the Linear differential equation, it turns out you can also solve it by using seperable equations. You can bring 6xv to the right side and factor 6x out...
Is there any practical application of this equation, or what is the meaning behind it? If it is just a mathematical problem-solving technique, spending too much time here is actually a waste of time. It would be great if you can tell me what problem can be solved
This type of equation randomly showed up during the solution steps for one of the example problems in my Applied Mathematics course about perturbation theory, and I had forgotten about it.
theyre named after him but what can these things model and are they autonomous or some shit?? examples of uses please or it wont stick...but i see its awesome
Dear Sir!! you are the one of the best professor I have ever seen. if u are on social media like Instagram or facebook then please let us know that so that we will start follow you on other platforms too.
Amazing, you've taught me all Cal classes + some stuff we didn't cover in my Differential Equations class (like in this video). A little comment I have is that I feel that you could've maybe done the work for all n to show how it always simplifies to a linear ODE. Keep up the good work!
Hey! I'm noticing that at 20:47, the bernoulli differential equation we obtained can be solved by separation. Is this correct or do we have to continue solving the DE using an integrating factor?
I have tried solving that bernoulli differential using the separation technique and obtained the same result! I answered my own question I guess... Would there be any reason why I can't do separation?
I love you. Thank you for being great. You make so happy every time because you explain everything so clearly. I hope you have a wonderful and fulfilling life that gives you everything you want Professor Leonard. You really do make a difference. Thank you. ❤
Thank you Prof Leonard! This class has been a lifesaver. Hoping you come back soon and give us a Linear Algebra course! Taking it next quarter and don't know how I'll make it without your help.
Just a suggestion why not divide everything by y^n then just substitute your v=y^(1-n) and dv/dx = (1-n)y^(-n) * dy/DC. That way you avoid having to do the multiplication and simplifying at the end. Maybe that can help :).
So somehow my university believes that the equivalent of 39 Prof Leonard videos on differential equations should be covered in 7 lectures.... Makes sense why I didn't understand it first time round....
The thing I can't "see" is why *minus* n? 1-n can be restated as +1-n or -n+1. If we just wanted a difference of 1 we could have used n+1 (or 1+n). Is it because when you differentiate a function of x^n you always get nx^n-1?
Long but useful and necessary. I tried learning this with quick 10 min videos but it wasn't cutting it. This is the kind of step by step explanation that works for complex topics.
As an alternative way to derive the substitution, you can divide both sides of the equation by y^n. Then the indicated substitution becomes easier to see.
It's good to practice the theory for linears, but like Bernoulli always gives a result, you can shortcut the linear to y = 1/rho • ∫ rho F(x) dx, or in this case v= and back-substitute. Just be careful to apply 1/rho (or 1/mu) to the entire integral including C.
I just spent 6 hours trying to work through this one math problem on homework not including the 4 hours spent the day prior. I looked at videos, notes, re-watched lectures, and asked peers, but I just was not getting anywhere. Finally, after all that stress and pulling my hair, I was able to follow the process, know what I was doing, and FINALLY get the correct answer. I cannot thank you enough. My grades are everything to me right now. Thank you.
Professor do you can divide videos and make playlist 1 and 2 for calculus 2 and 3? I need transcript because english my second language. I learning math and english langauge through your videos:-)) Thanks a lot!
Hi Prof Leonard, I have been watching your videos on DE. They are awesome. Especially the depth and the patience with which you have been explaining the specifics of each of the approaches. I am happy. Can you please share me your email id so that i could email you the specific DE or PDE model which would not have been covered in the current set of videos. In case you have already made the videos on my topics then you can guide me or you can recommend an alternative source. I am in the middle of the DE set of videos. Thank you
Thank you so much for the lectures. Do you think it would be easier to get the equation in standard form, then multiply through by the reciprocal of the y^n on the right side. Then substitute, that way you don't have to solve for y in the v=y term. Just take dv/dx and substitute on the left side dy/dx term.
Thanks for the video. However, I want to point out that at 8:47 you wrote y^2 = v^-1 => y = v^(-1/2). Correct me if I am wrong, but I think you should have written y = (+,-)v^(-1/2) instead. So I am not sure if you are allowed to make the following substitutions as you did.
I solved for y in the general equation, so I can simply plug in the numbers in the final equation and bypass most of the tedious algebra. xD Of course, I do regularly check so that I can solve these problems from scratch, maybe a couple times per month or something like that.
I’m not even in Diff Eq anymore, just stopping by to say hi lol. Sending good vibes from Florida!
bro this is A MILLION ZILLION TRILLION GAZILLION TIMES LESS CONFUSING THAN MY BOOK THANK YOU!!!!
Totally
I agree, but you could write this as a constant lol
Wow that’s a lot.
might as well send you my degree when i graduate since you're the only reason ill pass most of my courses lmao
🥰
For this presentation he's wearing a long-sleeved shirt without the sleeves rolled up. Good idea - this is serious subject matter, and we students can't be distracted by those biceps.
Thank you professor for the video. Looks like there is a typo at 24:05, Ce^(-2x) - 1, but you wrote + 1
He rectifies it later at 24:54
I just love you man
Youre the best maths teacher, would be a pleasure studying from you
This link th-cam.com/video/pSIqUnPIkP4/w-d-xo.html is also a nice video on Bernoulli's Equation.
Bernoulli is a name appeared very often in Fluid Mechanics.
I am an electrical engineering major and I can legitimately say you saved my life and kept me from having to repeat my Calc 4 class. Thank you so much for the explanations!
y'all call it calc 4, differential equations
@@abdallababikir9154 Yeah that's pretty weird. I've heard some schools mix linear algebra with ODE and call it Calc 4. Calc 4 at my school is just Vector Analysis.
im sorry Calc 4?
Didn't know Isaac Newton made a 4th sequal
@@warwick802 Calculus 4: Revenge of the Limit
I'm taking Calc3 and Diff Eq at the same time currently, and if it weren't for your videos, I know I'd be failing this year, because my professors are basically reading the textbook to us. I just wish I knew your channel existed when I took Calc1 and 2, I would have gotten better grades.
I'm sharing your channel with everyone I know.
Fan for life. Leonard is the GOAT
So I watched this video and the other homogeneous equation video yesterday night and this morning and was able to figure out 3 of my 4 exam problems (open notes). I want to thank you Professor Leonard for your videos because I would definitely have not been able to do my exam without your in depth explanations and tips 🙌🏽❤️
bruh I would kill for an open note DE exam
Please see my Chanel (Abdulhussein Alsultani) to see the differences between my method in solving th e examples)
Thank you very much.
I need this man leading me into combat
This link th-cam.com/video/pSIqUnPIkP4/w-d-xo.html is also a nice video on Bernoulli's Equation.
11:35 "that garbage" lmao i love this prof
Got a diff eq final tomorrow but I'll be fine... probably... maybe...
Trust yourself. You'll be fine
@@ProfessorLeonard Thank you, I will :)
@@ProfessorLeonard your videos + Slader.com have saved me and are allowing me to pursue a career that I never thought I would be able to because of the math requirements.
Two years late, but I also have a diff eq final tomorrow
@@aaron.glidden3yrs late but I don’t have diff eq final tmro
I scored pretty good marks in maths in my initial semesters because of him... when he had stopped making vedios i had no choice so left maths and took another subject as minor.. my syllabus for this sem was differential equations even though I dont have the subject anymore I still watch his vedios 😁😁
This link th-cam.com/video/pSIqUnPIkP4/w-d-xo.html is also a nice video on Bernoulli's Equation.
Professor Leonard, you are the reason why I will pass my Calculus 3 class and my Differential Equations Class. Thank you very Much! I wish my Professors were as good as you at explaining the work, you make math fun! I Wish I could have been in your classes.
Kind regards from South Africa.
Just a thought for the first problem at (6:24). If v = y^(-2), then dv/dx = -2y^(-3)dy/dx. If you divide the whole equation by y^3 you will be able to substitute in (-1/2)(dv/dx) directly. From there you can put it in linear form and solve. Saves you from all of the fraction exponents. Hard to explain in text but you essentially do two substitutions that gets you to dv/dx +2v= -2 in less steps. That's actually the brilliant thing about using substitution methods.
"leveraging the derivative of the power rule to undo a power rule inside of a product rule to undo that to find your original equation"= this video.
god, I freaking love you
This link th-cam.com/video/pSIqUnPIkP4/w-d-xo.html is also a nice video on Bernoulli's Equation.
These computations can be simplified. Indeed, set v=y^{-2}, you get that dv/dx=-2y^{-3}dy/dx. Then divide both sides of the equation dy/dx - y =y^3 by y^3. You get y^{-3} dy/dx -y^{-2}=1. So, (-1/2)dv/dx-v=1 or dv/dx+2v=-2 or dv/dx=-2v-2. This is a simple separable equation dv/(-2v-2)=dx. Now integrate both sides: (-1/2)ln(|v+1|)=x+c. This implies that ln(|v+1|)=-2x+C. Finally, apply the exponential and solve for v to get v=-1+Ae^{-2x}.
Played all your videos at the speed of 1.5 and really enjoyed your marvel. I absorbed everything you imparted like a sponge Professor Leonard. You have helped me get a straight A in Math courses. Keep up your excellent work, Professor!
Professor Leonard
- "He who is only an athlete is too crude, too vulgar, too much a savage. He who is only a scholar is too soft, too effeminate. The ideal citizen is the scholar athlete, the man of thought and the man of action."
~Plato
Even after I'm done with the diff eq class this semester, I will anxiously wait for the rest of your lecture videos. Hands down the best explanation anyone has given on Bernoulli's.
Yes..., yes you make sense!!!! The hand-waving is yet explained again!!! Thank you, Sir!!! This beats the heck out of a zoom white board where all we get is a non-rigorous proof (because coming from Calc III Students know what that is....not at all), of Bernoulli's DE's, then thrown into examples where the product rule is bypassed when we learned it, LIKE, yesterday. As a student, I do not need five shorthand examples in chicken-scratch on a digital whiteboard over zoom, I need 1 or 2 examples that dive into the details as you do in your videos. You're the Clark Kent of Mathematics, yet be the one who has found the Kryptonite of college paid professors that lack a basic understanding of how to actually teach what they understand themselves!
This is the most effective lecture on Bernoulli Differential Equations I have ever had.
I'm so so thankful for this really. My prof introduced diff eqs, its properties, separation of variables, linear DEs and integrating factors, and bernoulli's equation all within a span of 40 minutes and i could barely understand anything. You're a lifesaver
This helped me immensely. Thank you so much, sir!!
පට්ට...supiri..Thanks sir..respect ..
this absolute unit of a man is going to be the reason I pass my differential equation class god bless you good sir
Why make it simple, when you can do it the hard way...? Seriously Leonard, it took you 19 minutes (!) to state what you would obtain if you wrote dv/dx, and replaced dy/dx with the right-hand side of your the equation to be solved, instead of expressing y in terms of v (by means of a fct. that has 2 branches!)
Very helpful. With much gratitude Professor Leonard! You rock 🚀🎸🎯🏆
x y dx+(x^2-3y)dy=0
What's the answer
MasterClass!!!
I just wanted to mention that at 54:33, where you get the Linear differential equation, it turns out you can also solve it by using seperable equations. You can bring 6xv to the right side and factor 6x out...
Thank you so much sir ,,could'nt be more easier than this,,thanks alot,,❤❤❤love from Pakistan
Is there any practical application of this equation, or what is the meaning behind it? If it is just a mathematical problem-solving technique, spending too much time here is actually a waste of time. It would be great if you can tell me what problem can be solved
This type of equation randomly showed up during the solution steps for one of the example problems in my Applied Mathematics course about perturbation theory, and I had forgotten about it.
Okay, new example. So we have a differential equations BUUTTT it does not look that good. heard that like a dozen times now ;)
hello prof
can you please make a video on ricatti differential equations?
1st! Salute to Prof Leonard!
I'll second that! The master at work...incomparable.
theyre named after him but what can these things model and are they autonomous or some shit?? examples of uses please or it wont stick...but i see its awesome
This guy is a math rock be star. Where was he when I took Cal 1,2,3, and Diif EQ?
This man is far more handsome than a math professor has any right to be. (Also, he's really, really good at teaching.)
i have never ever a teacher like you after my school! you just awesome! thank you a lot!
Dear Sir!! you are the one of the best professor I have ever seen.
if u are on social media like Instagram or facebook then please let us know that so that we will start follow you on
other platforms too.
Has anyone told you , you look like superman? hahaha
The first equation can be done by variable separation right?
Amazing, you've taught me all Cal classes + some stuff we didn't cover in my Differential Equations class (like in this video). A little comment I have is that I feel that you could've maybe done the work for all n to show how it always simplifies to a linear ODE. Keep up the good work!
This link th-cam.com/video/pSIqUnPIkP4/w-d-xo.html is also a nice video on Bernoulli's Equation.
if nobody got me, i know that Professor Leonard got me. Amen
Awesome teacher!! Thanks for all the help!! =)
Just watched the whole thing and I finally understood Bernoulli!, thank you so much superman!
Can't believe I'm learning math from Clark Kent
you're unique and I feel you should one day be a professor at Carleton University or MIT .
Hey! I'm noticing that at 20:47, the bernoulli differential equation we obtained can be solved by separation. Is this correct or do we have to continue solving the DE using an integrating factor?
I have tried solving that bernoulli differential using the separation technique and obtained the same result! I answered my own question I guess... Would there be any reason why I can't do separation?
Your videos are amazing. Also the way you have placed the ads on you videos are also great.
I love you. Thank you for being great. You make so happy every time because you explain everything so clearly. I hope you have a wonderful and fulfilling life that gives you everything you want Professor Leonard. You really do make a difference. Thank you. ❤
Do you got Darth Vader behind the camera or am i goin crazy
thank you very much sir. love from sri lanka ❤
This video saved my life, thank you so much!!
i knew it........... only superman can save me now
I was so lost in class but this video helped me alot thanks
Thank you Prof Leonard! This class has been a lifesaver. Hoping you come back soon and give us a Linear Algebra course! Taking it next quarter and don't know how I'll make it without your help.
23:20
why is -1 changed to +1?
Yea i saw that too. I think its just a little mistake. He probably meant to write -1.
He fixed it if you continue.
Just a suggestion why not divide everything by y^n then just substitute your v=y^(1-n) and
dv/dx = (1-n)y^(-n) * dy/DC. That way you avoid having to do the multiplication and simplifying at the end. Maybe that can help :).
YOU ARE AMAZINGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
Was told about your channel today by a math major at my school. You're my hero.
So somehow my university believes that the equivalent of 39 Prof Leonard videos on differential equations should be covered in 7 lectures.... Makes sense why I didn't understand it first time round....
The first example was also a separable equation, -2 goes to the right side, you factor it out to get -2(1+v). Then just divide by 1+v and integrate
BERNOuLLI EQuATIoN>
I am a Mechanical Engineering major at UNF. I find your videos extremely helpful. I do tell everyone to watch your videos when I saw an opening
The thing I can't "see" is why *minus* n? 1-n can be restated as +1-n or -n+1. If we just wanted a difference of 1 we could have used n+1 (or 1+n). Is it because when you differentiate a function of x^n you always get nx^n-1?
Hey Professor, this comment is not related to this topic, I just wanted to ask you if you could make a video on chapter 3.8, the Newtons Method.
take a shot everytime he says cool
Long but useful and necessary. I tried learning this with quick 10 min videos but it wasn't cutting it. This is the kind of step by step explanation that works for complex topics.
sir thanks but which method are you using after obtaining the integrating factor when you are about to reach the step of substituting back V
you are the best thing to happen to our universe
As an alternative way to derive the substitution, you can divide both sides of the equation by y^n. Then the indicated substitution becomes easier to see.
It's good to practice the theory for linears, but like Bernoulli always gives a result, you can shortcut the linear to y = 1/rho • ∫ rho F(x) dx, or in this case v= and back-substitute. Just be careful to apply 1/rho (or 1/mu) to the entire integral including C.
I just spent 6 hours trying to work through this one math problem on homework not including the 4 hours spent the day prior. I looked at videos, notes, re-watched lectures, and asked peers, but I just was not getting anywhere. Finally, after all that stress and pulling my hair, I was able to follow the process, know what I was doing, and FINALLY get the correct answer. I cannot thank you enough. My grades are everything to me right now. Thank you.
Does anyone else hear someone breathing behind the camera? I am lowkey distracted by it
Sounds like you took too much addy
these videos saved my life...
At about 43minutes, why do you leave the Cx^4, instead of absorbing the x^4 into the C?
I can't tell the difference between black and purple. Maybe black and red, green, or blue?
I thought clark kent was a reporter
Professor do you can divide videos and make playlist 1 and 2 for calculus 2 and 3? I need transcript because english my second language. I learning math and english langauge through your videos:-)) Thanks a lot!
Hello is there a Differential Equations book that you would recommend?? Which one do you use when you teach?
This almost seems like cheating.
wish my tuition payed u not my current vegetable professors
2 hours recording this math video in his free time. Professor Leonard is a badass!
if every math teacher watched professor leonard the world would be a much better place
Hi Prof Leonard, I have been watching your videos on DE. They are awesome. Especially the depth and the patience with which you have been explaining the specifics of each of the approaches. I am happy.
Can you please share me your email id so that i could email you the specific DE or PDE model which would not have been covered in the current set of videos. In case you have already made the videos on my topics then you can guide me or you can recommend an alternative source. I am in the middle of the DE set of videos.
Thank you
Leonard, you're the best.
Thank you so much for the lectures. Do you think it would be easier to get the equation in standard form, then multiply through by the reciprocal of the y^n on the right side. Then substitute, that way you don't have to solve for y in the v=y term. Just take dv/dx and substitute on the left side dy/dx term.
Did you guys write the lecture or not? (In ur notebook), i want to write but it will take a lot of time.
Thanks for the video. However, I want to point out that at 8:47 you wrote y^2 = v^-1 => y = v^(-1/2). Correct me if I am wrong, but I think you should have written y = (+,-)v^(-1/2) instead. So I am not sure if you are allowed to make the following substitutions as you did.
the plus minus gets absorbed by C
اموت عليك
I solved for y in the general equation, so I can simply plug in the numbers in the final equation and bypass most of the tedious algebra. xD
Of course, I do regularly check so that I can solve these problems from scratch, maybe a couple times per month or something like that.
how do i comment with timestamp
Maybe it would be clearer to qualify the linearity, e.g. "linear in v".
Super helpful! Thanks so much for your detail!