That was an excellent video, and very timely. I am compleyely self taught, and while I have a basic knowledge of ODEs and PDEs, I am really addicted to complex analysis, and tend to focus on it at the expense of other topics. Your detailed, explicit, and thorough coverage of this topic is just what I needed. So, that is a rather wordy way of saying "thank you"!
@ProfessorMdoesScience9 ай бұрын
Really glad this was helpful!! :)
@oraange9 ай бұрын
This channel has a bright future!
@ProfessorMdoesScience9 ай бұрын
Thanks for your encouragement!!
@itsawonderfullife48029 ай бұрын
Bringing the same presentation excellence to math videos too. Many thanks.
@ProfessorMdoesScience9 ай бұрын
Glad you like the video!
@porit10239 ай бұрын
The mathematical justification part was really neat! Thank you so much for this video. Looking forward to the next one! :)
@ProfessorMdoesScience9 ай бұрын
Glad you liked that part :)
@materiasacra8 ай бұрын
I like the video. There is a slight problem with the integration constant c. Although no set of numbers is specified, it seems to be implied that everything is supposed to be real. At 9:38 we obtain c as exp(c_3), which for real c_3 implies that c is positive. At 12:03 you state that c can have 'any value', without discussion. This is actually true, but cannot be tacitly implied from what went before. At 13:42 you actually discuss negative values of c. This means that we are apparently not relying on the context in which the ODE arose, e.g. rabbit population in Australia, which might require y(x)>0 everywhere. Then you really need to argue explicitly that c can be non-positive.
@ProfessorMdoesScience8 ай бұрын
Thanks for the feedback! Fully agree with your comment, and we could certainly have been more careful in detailing this. Thanks again!
@angelmendez-rivera3519 ай бұрын
It is possible to be more rigorous in the procedure for solving the equation dy/dx = y. For example, one technically has to consider the possibility that y = 0. In this case, it is not possible to multiple both sides of the equation by 1/y, since 1/y does not exist. You would have to ask yourself, is it possible that y = 0? Well, if y = 0, then dy/dx = 0, and by transitivity, dy/dx = y, so yes, it is possible. You can set y = 0, and now restrict yourself to consider y ≠ 0, so now, it is meaningful to have (1/y)dy/dx = 1. When you antidifferentiate, though, you would need to be careful. For y < 0, the antiderivative of 1/y is ln(-y) + c0, while for y > 0, the antiderivative of 1/y is ln(y) + c1, and notice that c0 ≠ c1 is possible. Therefore, both of these cases must be accounted for, resulting in ln(-y(x)) + c0 = x + c2 for y < 0, and ln(y(x)) + c1 = x + c2 for y > 0. These can be rearranged to give y(x) = -(e^(c2 - c0))e^x and y(x) = (e^(c2 - c1))e^x respectively. We can simplify the results with the substitutions c3 = -e^(c2 - c0) and c4 = e^(c2 - c1). However, we need to be careful. e^(c2 - c0) > 0, regardless of whatever c0, c2 are, so c3 < 0 and c4 > 0. We have these restrictions for the possible values for c3, c4. Together with y(x) = 0, the equation is solved by y(x) = c3e^x, y(x) = 0, y(x) = c4e^x. Here, we can combine all three of the equations. Notice that 0 = 0e^x. Therefore, we can write y(x) = ce^x, where now, c can be any real number, and the three equations above are just the special cases where c < 0, c = 0, c > 0, respectively. This kind of rigor can be important when solving more complicated equations, where the constraints on the constants involved can actually be relevant.
@ProfessorMdoesScience9 ай бұрын
Absolutely! We always have to make choices as to what to include in a shortist video, and our approach is somewhat pragmatic as we say in the intro with "maths for science and engineering". Very often the functions and equations of interest in science and engineering are "well-behaved", and this principle is guiding our choices. But we are actually preparing problems and solutions to go with the videos, and in those we do explore some of these subtleties in detail, including exactly what you've described. We hope to release the problems+solutions soon. Overall, thanks for the feedback!
@richardthomas35779 ай бұрын
That is fantastic news about problems and solutions -- they will be especially helpful in the more advanced videos undoubtedly to come!! Thank you!@@ProfessorMdoesScience
@Adrian-wg7dz9 ай бұрын
Great video! Are you going to continue with the ones about two quantum state systems? Spin 1/2, Rabi oscillations...
@ProfessorMdoesScience9 ай бұрын
We are! We are currently working on a series on spin 1/2 on the QM front, and more to come after that too :)
@billbulgari9 ай бұрын
Thank you so much for the mathematical justification of the solution strategy at the end!
@ProfessorMdoesScience9 ай бұрын
Glad you find it useful! This is very often skipped, but we think it is best to properly justify all steps :)
@rsbenari9 ай бұрын
Brava. Clear, accessible -- without any loss of rigour. As always. Looking forward to this series -- and maybe some videos on Group Theory (tied back to your QM vids)? Thanks (so much!) for the channel.
@ProfessorMdoesScience9 ай бұрын
Glad you like it, and thanks for the suggestion! :)
@zacharyshifrel91079 ай бұрын
Great video as usual! I bet you’d both handle videos on QFT/renormalization/etc. really well. I’m about to get started doing some with manim.
@ProfessorMdoesScience9 ай бұрын
Good luck! And if we had the time we would like to cover all these topics and more, we'll see where we get... :)
@JohnAlexander-hj2nx9 ай бұрын
Hello! I'm happy that you are Back, i was watching your videos, i Made a question in your time evolution operator video, i hope You can respond me profesor M, thank you !!
@ProfessorMdoesScience9 ай бұрын
Just responded!
@JohnAlexander-hj2nx9 ай бұрын
@@ProfessorMdoesScience thanks but i don't understood a part can You respond me again pls
@Manish_18529 ай бұрын
You are pure gem... You make things so easy to understand If possible may you please make some videos on advanced quantum mechanics( Perturbation theory, variational methods, scattering etc.) your videos are very helpful thank you very much ❤
@ProfessorMdoesScience9 ай бұрын
Thanks for the suggestions! All of these topics are definitely on our to-do list!
@sandippaul4689 ай бұрын
I wished you guys taught me math and physics from high school
@ProfessorMdoesScience9 ай бұрын
Thanks for the kind words! :)
@amaljeevk39509 ай бұрын
Top class❤
@ProfessorMdoesScience9 ай бұрын
Glad you like it!
@paulbk23229 ай бұрын
Excellent, hats off to your wonderful efforts
@ProfessorMdoesScience9 ай бұрын
Thanks for your continued support!! :)
@snjy16198 ай бұрын
Today's session was very awesome And thankful to watch u live ❤😊😊 #Delhi University
@ProfessorMdoesScience8 ай бұрын
Glad you liked this. But what do you mean "watch live"?
@snjy16198 ай бұрын
@@ProfessorMdoesScience at delhi university
@ProfessorMdoesScience8 ай бұрын
@@snjy1619 Thanks for your answer! But still not sure I understand what you mean by "watch live"? That your professor played the video live in your lecture?
@snjy16198 ай бұрын
@@ProfessorMdoesScience I follow your lectures for QM n completed all the playlists Sir/ professor suggest me your videos ❤️
@snjy16198 ай бұрын
I request you Please upload more videos on QM 2
@زينالعابدينماجدمحمد9 ай бұрын
❤❤❤❤
@زينالعابدينماجدمحمد9 ай бұрын
❤❤❤❤❤
@suvrotica8 ай бұрын
Y should be e not 1 when C is 1 in the graph
@ProfessorMdoesScience8 ай бұрын
What are you refering to exactly? We only label the graph for the case c=0, we don't provide any labels when c=1.