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.
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.
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 🙌🏽❤️
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.
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.
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
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 😁😁
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.
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!
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!
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.
Today I did the second example by myself when a few days I had no clue what any of this was including a foggy idea on u-sub. It's crazy how many leaps in knowledge I'm experiencing from just following these videos. Thank you thank you thank you. Can never be overstated how much your help guides people like me who thought they were hopeless. You're the best!
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.
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.
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.
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. ❤
You are a seriously fantastic teacher! The detail and depth you provide with each new topic answers almost any question a student could possibly have. You also organize the information is such a way that it is easy to memorize the concept structure and problem solving structure.
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}.
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!
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.
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...
You are a freaking champ dude, I wish schools and textbooks taught like this. This was very clear and it helps a lot that you explain the logic behind the method.
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.
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.
@@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.
Just learned homogenous and Bernoulli equations a couple hours before my exam today using your videos, and I felt like I actually knew what I was doing on the exam. Thanks for being a great teacher!!
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.
My friend recommended this channel to me. It is really helping me study for my first DiffEQ exam. I did notice a POSSIBLE ERROR at 24:20 bottom right-hand corner. V=Ce^(-2x) -1 does not = Ce^(-2x) +1 Great video. Please keep making videos! This is crazy helpful!
Yep! I took a break for Thanksgiving and I'm going to take another for Christmas Break, but I will certainly be filming again in the Spring and until this project is complete. Enjoy!
Professor Leonard You are a living legend. Respect and Love, I have learned a lot from you Sir and still learning and will definitely donate in the near future cox I maybe getting a job :D.
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
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.
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.
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 :).
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?
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.
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
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.
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'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.
I just love you man
Youre the best maths teacher, would be a pleasure studying from you
This link ua-cam.com/video/pSIqUnPIkP4/v-deo.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.
Fan for life. Leonard is the GOAT
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
11:35 "that garbage" lmao i love this prof
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 ua-cam.com/video/pSIqUnPIkP4/v-deo.html is also a nice video on Bernoulli's Equation.
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
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.
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!
This is the most effective lecture on Bernoulli Differential Equations I have ever had.
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!
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.
Today I did the second example by myself when a few days I had no clue what any of this was including a foggy idea on u-sub. It's crazy how many leaps in knowledge I'm experiencing from just following these videos. Thank you thank you thank you. Can never be overstated how much your help guides people like me who thought they were hopeless. You're the best!
"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.
I need this man leading me into combat
This link ua-cam.com/video/pSIqUnPIkP4/v-deo.html is also a nice video on Bernoulli's Equation.
this absolute unit of a man is going to be the reason I pass my differential equation class god bless you good sir
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.
i have never ever a teacher like you after my school! you just awesome! thank you a lot!
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
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.
You have a gift for teaching and explaining complex topics. Thank you for providing these lectures for free to everyone
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.
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. ❤
You are a seriously fantastic teacher! The detail and depth you provide with each new topic answers almost any question a student could possibly have. You also organize the information is such a way that it is easy to memorize the concept structure and problem solving structure.
Bernoulli is a name appeared very often in Fluid Mechanics.
Was told about your channel today by a math major at my school. You're my hero.
this is the only one on youtube that doesnt make you remember the general solution. thank you
god, I freaking love you
This link ua-cam.com/video/pSIqUnPIkP4/v-deo.html is also a nice video on Bernoulli's Equation.
Very helpful. With much gratitude Professor Leonard! You rock 🚀🎸🎯🏆
Just watched the whole thing and I finally understood Bernoulli!, thank you so much superman!
Your videos are amazing. Also the way you have placed the ads on you videos are also great.
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}.
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 ua-cam.com/video/pSIqUnPIkP4/v-deo.html is also a nice video on Bernoulli's Equation.
This helped me immensely. Thank you so much, sir!!
Thank you so much sir ,,could'nt be more easier than this,,thanks alot,,❤❤❤love from Pakistan
you are the best thing to happen to our universe
Leonard, you're the best.
2 hours recording this math video in his free time. Professor Leonard is a badass!
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.
give this guy a raise
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...
This guy is a math rock be star. Where was he when I took Cal 1,2,3, and Diif EQ?
You are a freaking champ dude, I wish schools and textbooks taught like this. This was very clear and it helps a lot that you explain the logic behind the method.
I was so lost in class but this video helped me alot thanks
This video saved my life, thank you so much!!
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.
you're unique and I feel you should one day be a professor at Carleton University or MIT .
MasterClass!!!
thank you very much sir. love from sri lanka ❤
if every math teacher watched professor leonard the world would be a much better place
you made math look easy and cool for me, thank you for that.
these videos saved my life...
i love you professor leonard greetings from ecuador!
prof Leonard LIVING legend himself
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.
you are a life saverrrrrr
im gonna remember him in my prayers may god bless you
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
Just learned homogenous and Bernoulli equations a couple hours before my exam today using your videos, and I felt like I actually knew what I was doing on the exam. Thanks for being a great teacher!!
1st! Salute to Prof Leonard!
I'll second that! The master at work...incomparable.
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
Thank you pro.Leonard
පට්ට...supiri..Thanks sir..respect ..
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.
love you prof leonard
My friend recommended this channel to me. It is really helping me study for my first DiffEQ exam.
I did notice a POSSIBLE ERROR at 24:20 bottom right-hand corner. V=Ce^(-2x) -1 does not = Ce^(-2x) +1
Great video. Please keep making videos! This is crazy helpful!
You are the best, Sir. Loads of respect from Pakistan!
This link ua-cam.com/video/pSIqUnPIkP4/v-deo.html is also a nice video on Bernoulli's Equation.
@@jarvisty3640 subscribed :)
@@rainasajid6678 I found that video lecture so nice. There are a lot in the channel and I think is is just new.
Thanks professor for this fantastic lecture.
Super helpful! Thanks so much for your detail!
Thank you professor ! This helped me so much!
Thank you professor Leonard
Thank you Prof,, you are the best,, thank you very much
YOU ARE AMAZINGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
if nobody got me, i know that Professor Leonard got me. Amen
Awesome teacher!! Thanks for all the help!! =)
He’s back.
Yep! I took a break for Thanksgiving and I'm going to take another for Christmas Break, but I will certainly be filming again in the Spring and until this project is complete. Enjoy!
Professor Leonard You are a living legend. Respect and Love, I have learned a lot from you Sir and still learning and will definitely donate in the near future cox I maybe getting a job :D.
@@abdulmomun5592did you ever donate
thank you so much
Thank you!!!, these videos are really helping me understand the math behind the problems.
Umdah! Thanks alot Mr. Leonard
thanks king
Much love Professor!
This man is far more handsome than a math professor has any right to be. (Also, he's really, really good at teaching.)
You're the GOAT!
awesome work on explaining this
This link ua-cam.com/video/pSIqUnPIkP4/v-deo.html is also a nice video on Bernoulli's Equation.
Thanks❤
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
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.
Amazing! Thank you!
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.
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 :).
1:00:00
1:13:56
1:24:37
excellent vid!
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?