I have a degree in Mathematics gained as a mature student. I got a First. But my late father got frustrated with me when I belittled my own achievement. I said it wasn't a good course and they were not good teachers. The course was hard and I successfully jumped through their hoops. When I watch good tutors such as yourself, stress important points, and I recognise there importance, and how my course failed to emphasize it, and get me to internaise it, it makes me mad that people are ripped of by bad education and yet still have to pay their student loans. I don't wholly blame the lecturers, far from it, they're probably doing the best they can, under their own constraints, getting incoming students who have just been jumping hoops for other teachers who themselves are similarly constrained and betrayed themselves by the same education system that doesn't focus on understandings.
Yeah. You can earn a degree in a subject without _really_ knowing a lot about that subject. Teachers/professors make such a big difference -- for better or for worse.
After watching this video, I scrapped my entire lesson on 2nd Order DEs (first undergrad ODE class), and taught it in terms of Particular + Nullspace solutions, which really helped one student, who was the only one who had already taken Linear Algebra, and set everyone else up to understand L.A. better. Thank you.
Hi, I have just started learning PDEs from your videos. At least they are making some sense. In the other places they just start the with three basic forms. You have a great conversational style of teaching. Your students are lucky. Thanks.
Everything from this video seemed very simple and helped me understand how homogeneous solutions plus particular solutions relate to linear algebra! In the last 2 minutes I had to pause it and get up from excitement
23:31 Just to clarify, s is some solution of Ax = b and x_p is a particular solution. We can write s as s= x_p + (s - x_p) where s - x_p is in N, the nullspace of A. Thanks for proof.
Professor MathTheBeautiful, thank you for an excellent analysis on Linear ODEs and its overall importance in Ordinary and Partial Differential Equations. This is an error free video/lecture on RUclips.
You make Math Beautiful because you explain it so eloquently, in proper English, which I can’t enjoy enough of... Thank you for not teaching one without the other ;)
Excellent ! Thanks for your quality expression The way you explain subjects is really energetic and very clear, and you also deliver good hints from real world applications which further energizes the audience( at least me). For instance you included the way an expert in ODE investigates a real world problem and how it differs from how an expert in PDE tackles the problems. In fact there are much real world applications involving PDEs. You made clear at the beginning how the two worlds are
1:30 (1+t)u"-6u'+8u=r(t) I didn't actually tried to find the solution, but if I had to go about it with what I know so far, since it is a linear ode of 2nd order, I would solve it using power series and variation of parameters to solve the inhomogeneous part
@@harleyspeedthrust4013 Well, my DE course was pretty useless. I have no idea of what I was talking about. Completely forgot it in 4 years. I had to look it up to make sense of what I was talking about. Today my answer would be: "I would try to solve it symbolically with something like sympy and if that failed, I would do it numerically. Thats all I've got =P EDIT: Yeah, I don't know about the "variation of parameters" bit, but you can guess that a solution is A*e^(-5t) and solve a equation to figure out A (turns out A = 3/(25t+3)). That is a particular solution Yp. Then you use power series to solve the homogeneous equation and sum that solution to Yp. At least that part I didn't forget. Which is basically what this class was about: Any solution of a linear ODE is the sum of a particular solution (that you can hopefully guess, like in this case) and the solution for the homogeneous equation. Spoiler alert: it ain't pretty. Wolframalpha solves it in terms of Bessel function, which is the canonical solution for equations of the form t²u'' + tu' + (t² + z)u = 0, where z is a complex number. So the solution is non-analytic, as it is often the case with power series.
I already know all of this stuff, I had to learn by reading books and textbooks all formal stuff, this way of learning is way easier. I had to do it the hard way. I will complete the whole series because I admire good teachers.
23:35 - Quod Erat Demonstratum - "It was demonstrated," or "so demonstrated," approximately. Edit: Apparently I've been doing this wrong all along. It's "Q.E. Demonstrandum," or "It can be shown" / or the above translations are close enough. (ducks, while Latin scholars start throwing things)
? Matlab I think doesn't solve Ax=b always by Gauss elimination, it decides the solving method (QR, LU decomposition, .. etc) after checking the structure of the A matrix ?
I thought I understood this video until the last part where you concluded the null space of the differential equation must be 2 dimensional. Do the terms of the equation Au=b have dimension? I am not thinking of them as matrices, but as 2 functions (u and b) and a linear operator (A). So I don't see where the dimensional arguments from linear algebra enter the equation.
When you say that your particular solution plus a nullspace vector will give the "general" solution, is this equivalent to or a consequence of the ODE idea that the general solution will be equal to a linear combination of all of the unique solution vectors? (And hence why the homogeneous solution to the initial problem was c1*e^(2t) + c2*e^(4t))
I think your next video just answered the question in the affirmative (though maybe my question needs some refinement). Going to have to add this into my explanation for my ODE class (for the linear algebra graduates, especially).
Also when we are trying to draw an analogy from Matrix equations, what would be the dimenstions of A, u and b in Au=b equation ?. Is 3 e^(-5t) 1x1 or is it a two dimensional vector. Since Null space is two dinmensional , is the dimension of u = 2x1 . Dimension of A would be 1x2 ?
How do we know that the transformation is Linear. Is there is proof for the same ?. You have just said that ' one will gain an eye to concluding if the transformation is linear or not'. This transformation involves u'' and u', and I dont see how you saw that this transformation is necessarily linear Please guide ?
31:40 So there are no known ways of solving nonlinear equations? Shouldn't there be hordes of mathematicians trying to find some methods or a new theory of soling them then? :q Also, proving that there is a solution and that it has to be made of a certain number of pieces isn't very useful yet, unless you know how to actually find these pieces. Instead of coming up with a zoo of "special functions", I would be more comfortable if there was a general theory of finding these functions. A power series solution isn't very satisfying, since there's usually no way to go back from the power series to the original function even for quite simple functions. It's like getting the digits of a number but not knowing what that number actually is.
There is no general way to solve nonlinear ODEs analytically. In special cases there are some tricks that work, but generally it can't be done. However, all nonlinear ODEs can be solved numerically (i.e, with a computer program), and it's very easy to do so (he explains this in the previous video)
I have a degree in Mathematics gained as a mature student. I got a First. But my late father got frustrated with me when I belittled my own achievement. I said it wasn't a good course and they were not good teachers. The course was hard and I successfully jumped through their hoops. When I watch good tutors such as yourself, stress important points, and I recognise there importance, and how my course failed to emphasize it, and get me to internaise it, it makes me mad that people are ripped of by bad education and yet still have to pay their student loans.
I don't wholly blame the lecturers, far from it, they're probably doing the best they can, under their own constraints, getting incoming students who have just been jumping hoops for other teachers who themselves are similarly constrained and betrayed themselves by the same education system that doesn't focus on understandings.
Thank you for your comment. It is our hope that Lemma will make access to great teachers available to everyone.
Hythloday71 g
Yeah. You can earn a degree in a subject without _really_ knowing a lot about that subject. Teachers/professors make such a big difference -- for better or for worse.
After watching this video, I scrapped my entire lesson on 2nd Order DEs (first undergrad ODE class), and taught it in terms of Particular + Nullspace solutions, which really helped one student, who was the only one who had already taken Linear Algebra, and set everyone else up to understand L.A. better.
Thank you.
This is GOLD !!
The actual logical math instead of hitting your head with complex proofs and notation.
Thanks! That's actually a great way to describe my approach.
Hi, I have just started learning PDEs from your videos. At least they are making some sense. In the other places they just start the with three basic forms. You have a great conversational style of teaching. Your students are lucky. Thanks.
Hi Shwetank, thank you for your comment. I'm glad you're enjoying the videos.
It is surprisingly hard to find this kind of approach- honest, intelligent, rigorous, and refreshingly free from affectation. Very valuable.
Glad you liked it!
The final sentence of this video is incredible. Look within the problem as its presented and find the problem that is already solved.
Thank you!
Everything from this video seemed very simple and helped me understand how homogeneous solutions plus particular solutions relate to linear algebra! In the last 2 minutes I had to pause it and get up from excitement
23:31 Just to clarify, s is some solution of Ax = b and x_p is a particular solution.
We can write s as s= x_p + (s - x_p) where s - x_p is in N, the nullspace of A.
Thanks for proof.
Yes!
Professor MathTheBeautiful, thank you for an excellent analysis on Linear ODEs and its overall importance in Ordinary and Partial Differential Equations. This is an error free video/lecture on RUclips.
Lol "shut up and put it down. You don't know"😂😂😂
The best math teach on the Web
You make Math Beautiful because you explain it so eloquently, in proper English, which I can’t enjoy enough of... Thank you for not teaching one without the other ;)
Excellent !
Thanks for your quality expression
The way you explain subjects is really energetic and very clear, and you also deliver good hints from real world applications which further energizes the audience( at least me).
For instance you included the way an expert in ODE investigates a real world problem and how it differs from how an expert in PDE tackles the problems.
In fact there are much real world applications involving PDEs.
You made clear at the beginning how the two worlds are
Thank you!!! was very confused on particular solutions until I saw this video!
1:30 (1+t)u"-6u'+8u=r(t)
I didn't actually tried to find the solution, but if I had to go about it with what I know so far, since it is a linear ode of 2nd order, I would solve it using power series and variation of parameters to solve the inhomogeneous part
Yes, looks like a power series is a good way to solve that one
@@harleyspeedthrust4013 Well, my DE course was pretty useless. I have no idea of what I was talking about. Completely forgot it in 4 years. I had to look it up to make sense of what I was talking about.
Today my answer would be: "I would try to solve it symbolically with something like sympy and if that failed, I would do it numerically. Thats all I've got =P
EDIT: Yeah, I don't know about the "variation of parameters" bit, but you can guess that a solution is A*e^(-5t) and solve a equation to figure out A (turns out A = 3/(25t+3)). That is a particular solution Yp. Then you use power series to solve the homogeneous equation and sum that solution to Yp. At least that part I didn't forget. Which is basically what this class was about: Any solution of a linear ODE is the sum of a particular solution (that you can hopefully guess, like in this case) and the solution for the homogeneous equation.
Spoiler alert: it ain't pretty. Wolframalpha solves it in terms of Bessel function, which is the canonical solution for equations of the form t²u'' + tu' + (t² + z)u = 0, where z is a complex number. So the solution is non-analytic, as it is often the case with power series.
Awesome. Inspiring. Lucid. Energizing. A new fan!
Thank you and welcome aboard!
I already know all of this stuff, I had to learn by reading books and textbooks all formal stuff, this way of learning is way easier. I had to do it the hard way. I will complete the whole series because I admire good teachers.
Great lecture!
Thanks so much for your wonderful videos! I am actually understanding PDEs I think
Sir, Could you also prepare some videos on statistics? I a big fan of your way of teaching.
Sir , Could also please prepare videos on Calculus and Multi variable calculus ?
23:35 - Quod Erat Demonstratum - "It was demonstrated," or "so demonstrated," approximately.
Edit: Apparently I've been doing this wrong all along. It's "Q.E. Demonstrandum," or "It can be shown" / or the above translations are close enough. (ducks, while Latin scholars start throwing things)
Pavel for president...of the world
? Matlab I think doesn't solve Ax=b always by Gauss elimination, it decides the solving method (QR, LU decomposition, .. etc) after checking the structure of the A matrix ?
I thought I understood this video until the last part where you concluded the null space of the differential equation must be 2 dimensional. Do the terms of the equation Au=b have dimension? I am not thinking of them as matrices, but as 2 functions (u and b) and a linear operator (A). So I don't see where the dimensional arguments from linear algebra enter the equation.
I did not explain it in this video, but I believe I explain it in a later video
in the tranformation A u ...is u a scalar function described as a vector?? i have just started this coarse and this is where i am stuck,please help
When you say that your particular solution plus a nullspace vector will give the "general" solution, is this equivalent to or a consequence of the ODE idea that the general solution will be equal to a linear combination of all of the unique solution vectors? (And hence why the homogeneous solution to the initial problem was c1*e^(2t) + c2*e^(4t))
I think your next video just answered the question in the affirmative (though maybe my question needs some refinement). Going to have to add this into my explanation for my ODE class (for the linear algebra graduates, especially).
Also when we are trying to draw an analogy from Matrix equations, what would be the dimenstions of A, u and b in Au=b equation ?. Is 3 e^(-5t) 1x1 or is it a two dimensional vector. Since Null space is two dinmensional , is the dimension of u = 2x1 . Dimension of A would be 1x2 ?
I don't think all elements need to be part of the analogy. But whatever the dimensions of A are, its null space is two-dimensional.
26:42 OK, so how can we now express this ODE as the matrix A? :q
this bald gentleman saved another life.
How do we know that the transformation is Linear. Is there is proof for the same ?. You have just said that ' one will gain an eye to concluding if the transformation is linear or not'. This transformation involves u'' and u', and I dont see how you saw that this transformation is necessarily linear
Please guide ?
ruclips.net/video/lIwGFIFRspw/видео.html
@@MathTheBeautiful Thanks for this . It was helpful .
Still 2 dimensions even if we don't have constant coefficient?
Which part of the video are you referring to?
@@MathTheBeautiful 29:55
"Capital D is what I'd do to you"
I thought these lectures were for a university course, until you said "in high school, and then in Lemma". So who exactly are the audience?
The live audience is Drexel University students.
MathTheBeautiful Oh, so were you calling your Drexel University Linear Algebra class 'Lemma'?
flash 20:12
The flash meets the eraser.
26:16 Bruh...
And then people tell me we don't live in the Matrix ...
ILLUMINATING!
31:40 So there are no known ways of solving nonlinear equations?
Shouldn't there be hordes of mathematicians trying to find some methods or a new theory of soling them then? :q
Also, proving that there is a solution and that it has to be made of a certain number of pieces isn't very useful yet, unless you know how to actually find these pieces. Instead of coming up with a zoo of "special functions", I would be more comfortable if there was a general theory of finding these functions. A power series solution isn't very satisfying, since there's usually no way to go back from the power series to the original function even for quite simple functions. It's like getting the digits of a number but not knowing what that number actually is.
There is no general way to solve nonlinear ODEs analytically. In special cases there are some tricks that work, but generally it can't be done. However, all nonlinear ODEs can be solved numerically (i.e, with a computer program), and it's very easy to do so (he explains this in the previous video)