Hello from a math teacher in Pakistan. I am glad to see teachers taking initiatives and helping students in their problems. I am positive our videos are a great source of help for them. Good work
Haha Steve, i had exact the same "trouble" distinguishing between the 5 and the s back in the university...my workaround method was writing the s with some horns added in both its ends, you can't imagine how fancy they look...
Another way to solve the convolution of multiple trig functions: Based on the degree in the denominator for (s^2 + w^2)^n, the value of (n - 1) tells you how many times you'll eventually multiply trig by t. So you form a linear combination of t^k*sin(w*t) and t^k*cos(w*t), where w is the angular frequency, and k is a power that builds from 0 to (n-1). You then find corresponding Laplace transforms to each of these terms, and add up a linear combination with unknown coefficients, to equate to the original transform. Use the function parity property of convolution, you can eliminate half of the terms, and have half as many unknowns to solve for. f_odd(t) conv g_odd(t) = odd function f_even(t) conv g_even(t) = odd function f_odd(t) conv g_even(t) = even function If expecting odd functions, this means you can eliminate all t^even * cos(w*t) terms and t^odd * sin(w*t) terms. Vice versa, if you are expecting even functions. Then you proceed with solving for the unknown coefficients.
I have just found ur channel today and hands down ur already one of my favorite teachers on youtube. I wish i knew about u earlier. Ive been studying for some hours now and this is something i didnt do in a very long time. Your videos are very informative and very entertaining.
19 Here we can be tricky and build difference of squares from linear factor of denominator Then we will get constant term if we combine difference of squares with the other factor of denominator We will get 13=(s^2+9)-(s+2)(s-2) If we replace numerator by 1/13((s^2+9)-(s+2)(s-2)) we will have nice cancelling 20 16=s^4-(s^2-4)(s^2+4) and we have nice cancelling If we know hyperbolic functions we dont need partial fractions
This expression could have been written as 1/8[1/(s^2-4 -2) -1/(s^2+4)] and then 1/16[2/(s^2-4 -2) -2/(s^2+4)]. The Laplace Transform would , therefore be 1/16[Sinh(2t) -Sin(2t)], which is what Black Pen Red Pen got but in a convoluted way.
I've just finished the Laplace Transform Ultimate Study Guide video now I'm going to start watching this one, it's going to take me a while like the other one because I have other things to do.
Coming back and re-watching this video a couple years later, it occurs to me that on question 20 and other hard partial fraction decomposition problems the residue theorem from complex analysis can be used to help with it. You'd just have to calculate a couple derivatives for building up the powers of s, and the rest is fine.
I am taking differential equations in MIT and literally, you are saving my time with excellent exercises. Our book is just awful. Just imagine, some of your exercises appeared in my midterm exam
Number 16 no need to do that to find C and D You just have to multiply bt (s^2+4) both sides and then evaluate at s=2i It will follow -1/8 =C(2i) +D Immediately D=-1/8 and C=0 1/(s-2)(s+2)=1/(s^-4) evaluated at s=2i equals -1/8
For question 12 you can really easily simplify the partial fractions by letting some w = s^2 and then doing the partial fractions on w, and then later substituting s back in. edit: A simular trick can be used for Q16, where you can split up (s^4-16) into (s^2+4)(s^2-4) and again let w = s^2, do the partial fraction, reverse into the s world, then you can simplify it all down into 1/16(sinh(t) - sin(t))
Hii, thank you bprp for these marathon videos. It is very helpful even after 3 years and it will stay helpful. I would like to point that i couldn't open the file, which is not a big problem because we have the functions in the video and the description, but still it would be nicer to have them printed. Thank uu again
Residues are alternative way to partial fraction decomposition In fact complex partial fractions decomposition works better Residues are more comfortable also for inverse Z transform
It looks like the (6) case the cos(t-π/2) can be also sin(t)... In the (7) case cos(t)-sin(t) can be sqrt(2)*sin(π/4-t)... This should be possibly simplified in the t-domain. (10) delta(t-a) correct, but maybe then mistake before the inverse laplace if this is result. The delta is more practical in s-domain than in t-domain... (26) Try inverse laplace of s^2/(s^2+a^2)... How to make this? Why result is different for L^-1{1-a^2/(s^2+a^2)} versus L^-1{s/(s^2+i×a)} ∗ L^-1{s/(s^2-i×a)} (∗ = convolution)? Can this prove that convolution theorem and other inverse laplace can differ? Is correct answer -a×sin(a×t) ? Maybe is it possible to reconfigure the transformation to present the frequencies reference point at t=0- (zero minus) so that resulted delta(0) would be delta(0-) and then by using laplace validity for t>=0 removing this delta-function?
Thank you Mathematic Marathons GOD!!!, we appreciate your big brain, but there are a topic, Limits, can you give us a marathon about it? (I'm learn English, I speak spanish )
Dang I havent done laplace transforms since ODEs in my first year of uni. I graduated with an applied math bachelors back in 2016 but never had to do these again haha. Ive used the laplacian in PDEs many times but never these again😅. Blast from the past! Thanks for the video! Was able to do these since you mentioned LT is linear and with your note of what the laplace transform is its easy to go backwards Thank you!! Idk if Ill ever use this again (even in my masters when I start it) but it was fun to watch haha
Whenever I got a problem that stated "s" as a variable, my first line on my answer was "Let s = t" or something like that. S looks way too similar to 5 to be used a variable name. For the same reason, I have crossed hand-written every z for a very long time so that they don't look like 2.
Question: If there is 7.8 billion people on earth and we are all suppose to keep a 6 foot distance between ourselves to prevent the spread Cov-19, how much land do we need? It seems if we were to use a simple square grid we would need trillions of square miles. What would be the most efficient layout and is there enough land?
The most efficient layout would be a hexagonal close packing. Each person gets a hexagon to themselves, with a cross-flats distance (like the way you measure a wrench size) of 6 ft. To calculate the cross-points distance on a hexagon, given the cross-flats distance, you multiply by 2/sqrt(3). Let F equal the cross-flats distance, and P equal the cross-points distance. Such a hexagon is equal in area to 6 equilateral triangles of side length P/2, which each have an altitude of F/2. So the area of each triangle is: 1/8*P*F, and the hexagon area is 3/4*P*F. Plug in P=F*2/sqrt(3), and simplify, and we get: A_1hex = sqrt(3)/2*F^2 This means we'd need 31.177 square feet for each person. Multiply by 7.8 billion, and we get 2.43*10^11 ft^2 Divide by (5280 ft/mi)^2, and we get 8773 square miles required.
Do you have a whiteboard at your house? Great effort, I really do think you must have several clones who you swap in every 10min. Keep up the quality content!
Excellent question and barely no one asked. My first thing was NOT to have “math” in my channel name. Then I wanted to say something to intrigue other. Then I also wanted to point out the obvious thing that made me stand out. I think these were the things I was thinking about 10 years ago.
Laplace Transform Ultimate Study Guide: th-cam.com/video/ftnpM_RO0Jc/w-d-xo.html
Sir can you show us how we can calculate inverse Laplace transform by using integral relation
L^-1=1/2πi*integral(f(s).e^s*t.ds)
These marathon videos are becoming my most favourite thing to watch!
Thanks, glad to hear!!
same @svetozar
Hello from a math teacher in Pakistan. I am glad to see teachers taking initiatives and helping students in their problems. I am positive our videos are a great source of help for them. Good work
At first I thought you were gonna do the Laplace video but in reverse 😂
I should have just done that...
That would be a meta video lol.
The hell dude. I just started the original laplace marathon. And ALREADY?
😂
These marathons are great, your effort with the worksheet, timestamps, and everything else is greatly appreciated. Helped me out so much.
The beauty of all these videos is that you can watch again, again and again until you come to grasp the concept
All the s's are in red.
How do you distinguish your s and your 5?
blackpenredpen Can you do Fourier Transforms?
Not without a green pen.
Haha Steve, i had exact the same "trouble" distinguishing between the 5 and the s back in the university...my workaround method was writing the s with some horns added in both its ends, you can't imagine how fancy they look...
Can you tweet me a picture of how it looks like?
@@blackpenredpen don't have tw account, sent u to ur gmail instead
It would be nice a double and triple integrals marathon!!
I second this. Although maybe like 50 instead of 100.
also a fourier transformation series marathon.
yeah, fourier series marathon@@thecritiquer9407
Another way to solve the convolution of multiple trig functions:
Based on the degree in the denominator for (s^2 + w^2)^n, the value of (n - 1) tells you how many times you'll eventually multiply trig by t. So you form a linear combination of t^k*sin(w*t) and t^k*cos(w*t), where w is the angular frequency, and k is a power that builds from 0 to (n-1). You then find corresponding Laplace transforms to each of these terms, and add up a linear combination with unknown coefficients, to equate to the original transform.
Use the function parity property of convolution, you can eliminate half of the terms, and have half as many unknowns to solve for.
f_odd(t) conv g_odd(t) = odd function
f_even(t) conv g_even(t) = odd function
f_odd(t) conv g_even(t) = even function
If expecting odd functions, this means you can eliminate all t^even * cos(w*t) terms and t^odd * sin(w*t) terms. Vice versa, if you are expecting even functions. Then you proceed with solving for the unknown coefficients.
Finally!!!!!! Someone that understands, S and 5 can be really confusing especially if your handwriting is as bad as mine
Yeah, but why we use 's' not 'f' or 'g'?
@@FaranAiki Because the original inventor used s and f and g are reserved for functions.
you can use cursive s, you will no longer confuse them
I have just found ur channel today and hands down ur already one of my favorite teachers on youtube. I wish i knew about u earlier. Ive been studying for some hours now and this is something i didnt do in a very long time. Your videos are very informative and very entertaining.
19
Here we can be tricky and build difference of squares from linear factor of denominator
Then we will get constant term if we combine difference of squares with the other factor of denominator
We will get
13=(s^2+9)-(s+2)(s-2)
If we replace numerator by 1/13((s^2+9)-(s+2)(s-2)) we will have nice cancelling
20
16=s^4-(s^2-4)(s^2+4)
and we have nice cancelling
If we know hyperbolic functions we dont need partial fractions
Ahhhh so good!
Now we just need a fourier and inverse fourier transform marathon.
Yes, Please!
Wheres Fourier series and inverse Fourier series?
Exactly
This expression could have been written as 1/8[1/(s^2-4 -2) -1/(s^2+4)] and then 1/16[2/(s^2-4 -2) -2/(s^2+4)]. The Laplace Transform would , therefore be 1/16[Sinh(2t) -Sin(2t)], which is what Black Pen Red Pen got but in a convoluted way.
“Is this heaven?”
- “No this is a Inverse Laplace marathon”
“Hm, fair enough”
lol
Thank you thank you thank you! I would be lost in college without your videos!
I've just finished the Laplace Transform Ultimate Study Guide video now I'm going to start watching this one, it's going to take me a while like the other one because I have other things to do.
Can't I just play the other Laplace video in reverse? :-)
Hahaha that should work too!
😂😂 witty
It would sound wird
@@sgems13 thats the last of their worries lol
Coming back and re-watching this video a couple years later, it occurs to me that on question 20 and other hard partial fraction decomposition problems the residue theorem from complex analysis can be used to help with it. You'd just have to calculate a couple derivatives for building up the powers of s, and the rest is fine.
Maybe next could be some linear algebra videos? Ideas could be marathon on: finding Inverses, Eigenvectors, eigenvalues of matrices?
I am taking differential equations in MIT and literally, you are saving my time with excellent exercises. Our book is just awful. Just imagine, some of your exercises appeared in my midterm exam
I, once again, deeply thank you bprp!
This was EXTREMELY helpful!
❤
Number 16 no need to do that to find C and D
You just have to multiply bt (s^2+4) both sides and then evaluate at s=2i
It will follow
-1/8 =C(2i) +D
Immediately D=-1/8 and C=0
1/(s-2)(s+2)=1/(s^-4) evaluated at s=2i equals -1/8
Love the way you teach. Fast but informative.
I respect you so much. Right now I cant understand this topic, but I will comeback.
Do you guys know a marathon video of differential equations? I have to retake them for a subject and these videos are very useful
I really like these marathons.
Thanks!
Thanks blackpenredpen....take care and best wishes from Bangladesh 🇧🇩🇧🇩🇧🇩
For question 12 you can really easily simplify the partial fractions by letting some w = s^2 and then doing the partial fractions on w, and then later substituting s back in.
edit: A simular trick can be used for Q16, where you can split up (s^4-16) into (s^2+4)(s^2-4) and again let w = s^2, do the partial fraction, reverse into the s world, then you can simplify it all down into 1/16(sinh(t) - sin(t))
Sir i appreciate you. You are the best!
Greetings from Turkey.
Hii, thank you bprp for these marathon videos. It is very helpful even after 3 years and it will stay helpful.
I would like to point that i couldn't open the file, which is not a big problem because we have the functions in the video and the description, but still it would be nicer to have them printed. Thank uu again
incredible was able to find error in the work we did thanks so much.
Solution of partial differential eq using Laplace and Fourier transtorm
You are a great teacher!❤❤❤
Residues are alternative way to partial fraction decomposition
In fact complex partial fractions decomposition works better
Residues are more comfortable also for inverse Z transform
Thank you, this quarantine has led me to study differential equations on my own, thanks from Honduras
An inverse Laplace marathon? Man, there must be a lock-down! 😂
Example (16) A=1/4 ; B=-1/4 ; C=0 ; D=-1 . Thanks !!!
You are the best, thank you so much :D
Video still useful today. Thanks teacher!
But @blackpenredpen Can you also do fourier transform please?
A higher order differential equations marathon would make a complete set!
It looks like the (6) case the cos(t-π/2) can be also sin(t)...
In the (7) case cos(t)-sin(t) can be sqrt(2)*sin(π/4-t)...
This should be possibly simplified in the t-domain.
(10) delta(t-a) correct, but maybe then mistake before the inverse laplace if this is result. The delta is more practical in s-domain than in t-domain...
(26) Try inverse laplace of s^2/(s^2+a^2)... How to make this?
Why result is different for L^-1{1-a^2/(s^2+a^2)} versus L^-1{s/(s^2+i×a)} ∗ L^-1{s/(s^2-i×a)} (∗ = convolution)?
Can this prove that convolution theorem and other inverse laplace can differ? Is correct answer -a×sin(a×t) ?
Maybe is it possible to reconfigure the transformation to present the frequencies reference point at t=0- (zero minus) so that resulted delta(0) would be delta(0-) and then by using laplace validity for t>=0 removing this delta-function?
You are great I like this channel more than other ...
I was additcted to marathon's race (done 3) now I am addict to your marathon 👍
From Italy with love!
Exciting!!
You're a champ! This helped a ton. Thanks!
My favorite teacher
I love u sirrrrrr❤❤i love the way u teach us.its easy to understand
Well done !! I gotta ask a question : for Q15, F(S) doesn't converge to 0 when s goes to infinity, therefore we can't use differentiation, right ?
oh srry, right ln(1)=0. My bad.
Awesome working through the struggle
hello, thanks for the videos! Did you do any via the Fourier Transform? Something similar to the Laplace?
thank you
Next Video: Differential Equation 2nd Order (btw. keep up with the great content, love it!)
THANK YOU SO MUCH
For #16, the result could have been written as (1/16)[sinh(2t)-sin(2t)], which makes more sense to me.
In minite 48 your solution Is correct
But i solve it by adding (s²+1-s²) twic
I love the beginning, I also mess up my 5's and s's
Thank you Mathematic Marathons GOD!!!, we appreciate your big brain, but there are a topic, Limits, can you give us a marathon about it? (I'm learn English, I speak spanish )
Your video was badly needed! Most books and sources just blow through one example…forgetting that perfect practice makes perfect! Thanks!
Glad to help 😃
Thanks!
*Will you make a video with 100 limits ?*
Amazing & thanks
this is a good video for me to practice my engineering math ：))
aqui, en pleno verano practicando, para no olvidarse de esto... un cracj bprp... saludos :)
Hey can u do a marathon on z transform and inverse z transform
Getting bored with the quarantine bprp? Ye, me neither
Everyone's saying that they're all bored while we're just chilling at home doing maths
can't imagine that this was posted 9 months ago now..
@@aleks456 I can't handle this anymore, I've passed calc 2 and 3 since then but this is enough
@@Marcox385 Same bro. Let's just hold on and wait for this to finish!
Thanks for keeping my quarantine filled. Love from India😍
Lg /amazing video's
1:08:43 i realised it is possible to use cover up by substituting s^2=u
and A and C is automatically 0
AMAZING AS ALWAYS
José Julián Maldonado Camacho indeed
thank you so much professor can you do the same with fourier transform ?please
Thank you very much Steve...continuous..we are all love you ❤❤❤
خلف ابراهيم no doubt
Thanks!!!
I feel so unlucky not being your student. Your teaching is eloquent ☺️
According to f(t-a)*u(t-a) = iLapace{e^(-as)F(s)} , for a=0, all inverse Laplace transforms should be multiplied by u(t), am I right?
你回来了曹老师👍
Thanks!!
Who's teacher cao? Is that bprp?
thanks ... is there a marathon for fourier ?
We have to call up dr. Peyam for this! Lol
@@blackpenredpen is it really harder than Laplace and it's inverse and power series and the all mighty integral Marathon !??
Good morning sir
You got some stamina!
I love you. thank you.
2:02:17 and 2:02:31 reveal that an infinitesimal amount of exhaustion has affected the dual-pen-wielding Jedi master.
OMG Love this video
Can you do video about the Fourier transform?
May I ask you if you can make physics video? Thanks a lot
Marathon for integration with residues?
That face expression switch at 34:45
Dang I havent done laplace transforms since ODEs in my first year of uni. I graduated with an applied math bachelors back in 2016 but never had to do these again haha. Ive used the laplacian in PDEs many times but never these again😅. Blast from the past! Thanks for the video! Was able to do these since you mentioned LT is linear and with your note of what the laplace transform is its easy to go backwards
Thank you!! Idk if Ill ever use this again (even in my masters when I start it) but it was fun to watch haha
Whenever I got a problem that stated "s" as a variable, my first line on my answer was "Let s = t" or something like that. S looks way too similar to 5 to be used a variable name. For the same reason, I have crossed hand-written every z for a very long time so that they don't look like 2.
Write a cursive s, to tell it apart from a 5.
That was too much fun . Isn't it
Mr.bprb thanks for your effort.....I wanna tell you that you haven't put negative sign before the term tsin(4t) in no.22
Next up : W Lambert Marathon
Great!
Sir super vedio I am waiting
Question: If there is 7.8 billion people on earth and we are all suppose to keep a 6 foot distance between ourselves to prevent the spread Cov-19, how much land do we need? It seems if we were to use a simple square grid we would need trillions of square miles. What would be the most efficient layout and is there enough land?
The most efficient layout would be a hexagonal close packing. Each person gets a hexagon to themselves, with a cross-flats distance (like the way you measure a wrench size) of 6 ft. To calculate the cross-points distance on a hexagon, given the cross-flats distance, you multiply by 2/sqrt(3). Let F equal the cross-flats distance, and P equal the cross-points distance.
Such a hexagon is equal in area to 6 equilateral triangles of side length P/2, which each have an altitude of F/2. So the area of each triangle is: 1/8*P*F, and the hexagon area is 3/4*P*F. Plug in P=F*2/sqrt(3), and simplify, and we get:
A_1hex = sqrt(3)/2*F^2
This means we'd need 31.177 square feet for each person.
Multiply by 7.8 billion, and we get 2.43*10^11 ft^2
Divide by (5280 ft/mi)^2, and we get 8773 square miles required.
So is the hexagon the most efficient way because it tessalates? @@carultch
sir thank you very much for your greatest efforts sir.........❤❤❤
😅
I am using this marathon for exam preparation 😂 thanks btw 😁👍
Wonderful 😊
on problem 8... why not add and substract s^2 on top?
Do you have a whiteboard at your house? Great effort, I really do think you must have several clones who you swap in every 10min. Keep up the quality content!
I wonder how you derived the creativity to decide you YT name?!
Excellent question and barely no one asked.
My first thing was NOT to have “math” in my channel name. Then I wanted to say something to intrigue other. Then I also wanted to point out the obvious thing that made me stand out. I think these were the things I was thinking about 10 years ago.
i would totally give a like to tell u how helpful u are ♥️