i would've never passed calc 2 without you, and now i'm going to pass differential equations with an A also thanks to you. i can never thank you enough for what you do here. ur the best
it's easy because the problem is simplified. In reality, though, the differential equations are much more than constant rates and/or the solutions are not separable, solvable analytically and have to use numerical methods (or a calculator) to solve for. (if the in and out rates change you get an inhomogeneous differential equation)
Compare the elaborate detailed humble professor Leonard with some arrogant fidgy brief confusing Ivy League professors on the same subject and you will realize that quality education or teaching is anything but making sure your students get it. Professor Leonard understands his subjects and knows how to explain it to the average student. He teaches from the bottom of his heart. He is not there to convince you that you are dumb enough to drop the course like some egoistic professors! Thanks a ton to Professor Leonard. I wish him long happy life!
I was out sick last week so I completely missed the lecture for this. I tried reading the book and was SO lost. You have saved me sir. IDK what I would have done without this!
You genuinely just saved my grade. I’ve been struggling with mixing and modeling since the topic was introduced. The textbook I have is really difficult to understand, but this video was so easy to follow along.
Thank you, sir. My prof for this class jumps to the equations and just says to plug stuff in which is quite useless. You dive into the deeper meaning of pure math. Thank you.
Check out "The Book of Proof" by Richard Hammack. There's a PDF online that the publisher posted to make it freely available (so it's legal). That saved me in discrete.
Hi, thanks for the video, it's great as usual. But you should try to add a more descriptive information to this video. So you can get more hits when people search for it!
I switched to a Dual Major in Computer Science and Mathematics because of you. Classes I need you to teach please: Advanced Calculus I & II, Differential Equations II, Number Theory, Introduction to Proofs, and Statistics II.
Thank you so much for making this class for us. I'm taking this class and Linear algebra. I wish you had a Linear algebra course because you've basically been my college math professor for the all other maths that I've taken and I'm struggling without you in Linear.
Very nice explanations! I was wondering: Can these problems also be solved via separation of variables by just dividing both sides by that # + #x then multiplying both sides by dt, leaving you with an expression 1/(# + #x) dx = 1 dt, then integration both sides, using a quick U-substitution for the left side & getting t + C on the right side?
I hope you found the answer already, but it is possible if the separable method can be set up. For examples one and two separating and integrating would give the same function x, but I believe example 3 is not separable and should be solved using a method like the one in the lecture.
Prof Leonard, at 29:52 I do understand that you are going to use an integrating factor but this scenario is a separable D.E. so were you trying to use integrating factors for extra practice?
My brain just had a system failure when I saw the units in the last example ^^ the ones before I could cope with, but in the last one the 50 lb in 100 galons just killed me
It would be nice to include in your last exercise (or a new exercise ?) how the outflow changes with respect to the volume. The more volume the faster the water flows out. (or isn't that possible?) b.t.w. I enjoy the lessons. :-)
Amazing lesson as usual! Your explanations are really clear and easy to understand. Makes math fun. One thing i want to add, i think there was a mistake around 58:45 while taking the integration. Shouldnt the int. factor be (50+t)^3/2. Didnt effect the result though because it was a constant as I learned from the previous videos :)
no, nothing you said made any sense. 100+2t is u, he simply subbed the u back in. the 2 he divided by was outside the bracket, thats why it was 3/2 instead of 3
@@AB-gu9ui Food for thought. Step one, factorise (100+2t) as 2(50+t). Step 2, take 3/2 outside the integral. Let U=(50+t) and the integral becomes 3/2*ln(50+t). Can both answers be correct?
For the Chlorine concentration problem, why is it x/8000 for the concentration of the chlorine in the pool? The units doesn't work out because the m^3 cancels out, unlike in the first term, where you would get m^3/day for the amount of chlorine coming in. I think dx/dt should just be 500 * 0.05 - 500 x? Where am I wrong here?
We are pouring in pure water and not brine so the 100kg of sugar already in the tank should be accounted for. How will we get the 10kg at a certain time when you assumed the tank has no sugar? Where will the 10kg come from since we pouring in pure water and pouring out pure water because the 100kg of sugar was ignored?
I believe the concentration OUT = concentration OUT, if you don't have a filter on the OUT to reduce the concentration OUT. This is because the amount of salt in the container increases as the volume increase, proportionally
Prof, I'm surprised nobody mentioned this, but you made a grave error in the brine example at the end. You integrated 3/(100+2t) to be 3/2 * ln(100+2t) whereas it should've been 3/2 * ln(50+t)
He used integrating factors, which some function 'we call it p(x)' is equal to e to the power of (integral of the number by x OR y) (not the derivative) then we solve that. If that's confusing which it probably is then think of it this way instead: We are basically trying to manipulate the problem by finding something with what we have and then multiplying it to all the terms, Then we can do what we would call 'integrating factor' which basically is a DERIVATIVE product rule BACKWARDS like this, it turns this: dx/dt * (Number1) + (Number2) * x = (Number3) INTO THIS: d/dx(x*(something)) Sorry I can't explain it exactly but lookup his video on (Integrating Factors)
Yes, some can certainly be done that way. But, the more challenging one's are typically linear. So, to prepare for those, I like to show the easier ones in the technique that will be used more often. Good catch though!!!!
Hello professor, Last year i decided to start studying again and get my BA. But i was truly worried because, i did’t study math since 2006.eventhough english is not my first language l was able to pass calculus depending on your videos. And this year i am studding Differntial equations and i got B+ on my midterm exam. Thank you for making every thing really easy for me.😭
@@infernape716 I was actually quite confused as to what x(t) really represented. I thought that it was the amount of solution, like brine or water with sugar. Apparently, it just represents the solute
maybe there is something wrong with my headphones but i swear it sounds like there is a hospital ventilator in the background... weird! Otherwise great video!
i would've never passed calc 2 without you, and now i'm going to pass differential equations with an A also thanks to you. i can never thank you enough for what you do here. ur the best
YA' but sometimes there not all complete.😁.
is it just me or is this some how easy but really hard at the same time ?
it's easy because the problem is simplified. In reality, though, the differential equations are much more than constant rates and/or the solutions are not separable, solvable analytically and have to use numerical methods (or a calculator) to solve for. (if the in and out rates change you get an inhomogeneous differential equation)
@@duckymomo7935 you killed me lol
Welcome to calculus
its eas because its a lot of algebra rules with some derivatives but there is just a lot in diffeq's
Agreed.
Are you planning to make videos about Discrete Mathematics or Linear algebra? It would be so great
So impresive that you make everything look like grade 1 math, great job professor !
Wow. Now I don't need Jim to help me in my DE classes. Thank you, professor!
Compare the elaborate detailed humble professor Leonard with some arrogant fidgy brief confusing Ivy League professors on the same subject and you will realize that quality education or teaching is anything but making sure your students get it. Professor Leonard understands his subjects and knows how to explain it to the average student. He teaches from the bottom of his heart. He is not there to convince you that you are dumb enough to drop the course like some egoistic professors! Thanks a ton to Professor Leonard. I wish him long happy life!
You are the mvp man we don’t have time in class to talk about all this. So helpful.
Even though I am super busy I still find time for these awesome lectures. Thank you Professor Leonard.
But seriously, [23:35], Jim needs to be fired.
I was out sick last week so I completely missed the lecture for this. I tried reading the book and was SO lost. You have saved me sir. IDK what I would have done without this!
This man deserves every follow he gets. Fantastic teacher!
Outstanding! I found the integrating factor concept very difficult, but the repeated examples finally made it all clear.
You genuinely just saved my grade. I’ve been struggling with mixing and modeling since the topic was introduced. The textbook I have is really difficult to understand, but this video was so easy to follow along.
Was sceptical about watching an hour long video just to understand something... BUT MAN I UNDERSTOOD IT SO WELL. Thanks professor 👍
Thank you for doing what you do and posting these videos for everybody to see. You deserve more recognition
Thank you professor, you made it silky smooth now
My professor didn’t post an example problem on these, I’m glad I found this video! Thank you so much
Thank you, sir. My prof for this class jumps to the equations and just says to plug stuff in which is quite useless. You dive into the deeper meaning of pure math. Thank you.
I have been struggling with that third example for 2 days! Thank you!!!!
You are a genius! Thank you for your great lecture I watched it twice already and it makes sense. Keep doing you!
I have a quiz on this exact thing. I really struggle with this type of problem and this helped. I love the in depth structure of the video.
Thank you for this video. I cannot emphasize how well this is explained.
i legit started doing this today and you uploaded as i started doing the homework, you are the man!
You are honestly a life savior I would have failed math ages ago without you
thank you from the bottom of my heart- just aced my quiz!
Hey Leonard! You're the man. Great video as always
eye opening! great lecture!
this video saved my life i love you
Love all of your math videos.....your great at make something that is hard very easy!!!
i've been waiting for this topic from u prof,thanks.
Thank you very much prof leonard
Thank you for the complete step-by-step walkthrough ^__^
Always remember a good teacher , he was a good student
Brilliant lessons! Thanks!
What my professor cares is to show how smart he is by skipping some explanations.
This video is amazing, thank you
Thank you professor!
Great examples.
A lovely lesson! Thanks!
this is amazing.. thank you so much
Thank you very much sir, Really helpful!
That was really helpful.Thanks a bunch
Prof you are great.
Well done! Thank you so much!
If only u make a course about Discrete Mathematics 💔
Check out "The Book of Proof" by Richard Hammack. There's a PDF online that the publisher posted to make it freely available (so it's legal). That saved me in discrete.
Emerald905 thank you!
Lifesaver leonard!
thanks Superman!!
Thank you so much.
Misss your lectures ❤
Hi, thanks for the video, it's great as usual. But you should try to add a more descriptive information to this video. So you can get more hits when people search for it!
THANK YOU
you are the man
I switched to a Dual Major in Computer Science and Mathematics because of you. Classes I need you to teach please: Advanced Calculus I & II, Differential Equations II, Number Theory, Introduction to Proofs, and Statistics II.
Where is the Nobel prize for this guy?
i like this and i am subscribing
thanks a lot !
Thank you sir.
Finally got it!
Thanks a lot!!
Thank you so much for making this class for us. I'm taking this class and Linear algebra. I wish you had a Linear algebra course because you've basically been my college math professor for the all other maths that I've taken and I'm struggling without you in Linear.
how did linear go were you able to find any other resources??? I'm actually taking Diff Eq. with linear algebra this semester.
@@loganatkinson2043 czcams.com/play/PLHXZ9OQGMqxde-SlgmWlCmNHroIWtujBw.html
This guy was pretty good
@@loganatkinson2043 you dont feel old till you have stop watching profesor leonard
cuz of no more videos :(
Very nice explanations! I was wondering: Can these problems also be solved via separation of variables by just dividing both sides by that # + #x then multiplying both sides by dt, leaving you with an expression 1/(# + #x) dx = 1 dt, then integration both sides, using a quick U-substitution for the left side & getting t + C on the right side?
I hope you found the answer already, but it is possible if the separable method can be set up. For examples one and two separating and integrating would give the same function x, but I believe example 3 is not separable and should be solved using a method like the one in the lecture.
Prof Leonard, at 29:52 I do understand that you are going to use an integrating factor but this scenario is a separable D.E. so were you trying to use integrating factors for extra practice?
because this is application of linear differential equations
My brain just had a system failure when I saw the units in the last example ^^ the ones before I could cope with, but in the last one the 50 lb in 100 galons just killed me
thank youuuuu sooooo muchhh
I used separable equations to solve the first problem n it worked it there a reason u used linear
It would be nice to include in your last exercise (or a new exercise ?) how the outflow changes with respect to the volume. The more volume the faster the water flows out. (or isn't that possible?) b.t.w. I enjoy the lessons. :-)
Amazing lesson as usual! Your explanations are really clear and easy to understand. Makes math fun.
One thing i want to add, i think there was a mistake around 58:45 while taking the integration. Shouldnt the int. factor be (50+t)^3/2. Didnt effect the result though because it was a constant as I learned from the previous videos :)
u r right, he made a mistake.
no, nothing you said made any sense. 100+2t is u, he simply subbed the u back in. the 2 he divided by was outside the bracket, thats why it was 3/2 instead of 3
@@AB-gu9ui Food for thought. Step one, factorise (100+2t) as 2(50+t). Step 2, take 3/2 outside the integral. Let U=(50+t) and the integral becomes 3/2*ln(50+t). Can both answers be correct?
Great job as usual. I am just curious, do you plan any other advanced math courses in the future?
Perhaps. I'm thinking maybe linear algebra someday.
Pls do lol I'm taking it this semester
@@ProfessorLeonard PLEASE DO A SERIES ON LINEAR ALGEBRA MY NEXT SEM HAS IT PLSPLS
For the Chlorine concentration problem, why is it x/8000 for the concentration of the chlorine in the pool? The units doesn't work out because the m^3 cancels out, unlike in the first term, where you would get m^3/day for the amount of chlorine coming in. I think dx/dt should just be 500 * 0.05 - 500 x? Where am I wrong here?
We are pouring in pure water and not brine so the 100kg of sugar already in the tank should be accounted for. How will we get the 10kg at a certain time when you assumed the tank has no sugar? Where will the 10kg come from since we pouring in pure water and pouring out pure water because the 100kg of sugar was ignored?
59:14 Isn't that suppose to be e^[ (3/2) * [ ln(50 + t) ] ]professor? Thank you for the video by the way.👍
For problem one, when he got P(t)=e^t/200, how come he didn’t put an ln in front of the e to cancel out the e and be left with t/200 as P(t)?
Hi professor, In the 2nd example, Ci was 5% . why you didn't multiply it with the adding rate 500 Ft^3?
he did, that's how he got 25.
I believe the concentration OUT = concentration OUT, if you don't have a filter on the OUT to reduce the concentration OUT. This is because the amount of salt in the container increases as the volume increase, proportionally
Prof, I'm surprised nobody mentioned this, but you made a grave error in the brine example at the end. You integrated 3/(100+2t) to be 3/2 * ln(100+2t) whereas it should've been 3/2 * ln(50+t)
South Park Memberberries reference @17:13
i died ahhahhaa
glad i wasnt the only one that noticed
The tank originally has had 100 gallons of the solution in it, why is it 300/(5-2), not (300-100)/(5-2)? 54:00
isn't it supposed to be 3t instead of 3^t?
58:30 where did e come from? and what happened to the rest of the equation?
He used integrating factors, which some function 'we call it p(x)' is equal to e to the power of (integral of the number by x OR y) (not the derivative) then we solve that.
If that's confusing which it probably is then think of it this way instead:
We are basically trying to manipulate the problem by finding something with what we have and then multiplying it to all the terms,
Then we can do what we would call 'integrating factor' which basically is a DERIVATIVE product rule BACKWARDS
like this, it turns this:
dx/dt * (Number1) + (Number2) * x = (Number3)
INTO THIS:
d/dx(x*(something))
Sorry I can't explain it exactly but lookup his video on (Integrating Factors)
Refer to lecture 15 of his differential equations playlist.
Darn it Jim!
You must be the Super-man. You probably have an external hard drive hidden somewhere on you.
isnt the C's value for the last problem -95000 instead of -5000
@29:00 couldn't you have just done that problem using separable?
Yes, some can certainly be done that way. But, the more challenging one's are typically linear. So, to prepare for those, I like to show the easier ones in the technique that will be used more often. Good catch though!!!!
CIRRRCUUUUIITSSSSS PLEASE
RIP Jim
Man he really has a problem with this Jim guy.
Hello professor,
Last year i decided to start studying again and get my BA. But i was truly worried because, i did’t study math since 2006.eventhough english is not my first language l was able to pass calculus depending on your videos. And this year i am studding Differntial equations and i got B+ on my midterm exam.
Thank you for making every thing really easy for me.😭
Great job on that B+!!!
Gangadar hi Shaktimaan h.
Damn it, Jim, I'm a doctor not a mixologist.
member limits? i member.
I know you're a math professor, but you can do a course in statics or dynamics?
Sorry, how about Complex or Real Analysis?
the next thing he tackles is probably linear algebra.
If this was chemistry, the 'amount' and 'concentration' would be kinda different hahaha
yup, the amount of solution is really amount of solute and we would measure the concentration in mol/L not kg/L
@@infernape716 I was actually quite confused as to what x(t) really represented. I thought that it was the amount of solution, like brine or water with sugar. Apparently, it just represents the solute
Clark Kent, is that you?
3 jealous math teachers disliked this video
Mathgod
i hope diffusion fick gonna be created
Is that Clark kent?🤨
Hello Clark Kent
maybe there is something wrong with my headphones but i swear it sounds like there is a hospital ventilator in the background... weird! Otherwise great video!
Yeah, it's the noise cancelling I had to use back then... made a weird sound for some people. Sorry!
@@ProfessorLeonard so im not crazy. Ty:)
clutch
This dude is such a chad lol