can y'=y^i?
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- Опубликовано: 12 сен 2024
- We will solve an imaginary differential equation dy/dx=y^i for fun! We will need to use the reverse power rule, Euler's formula for complex exponential, writing a complex number in standard form and the formula for (a+bi)^(c+di): • the tetration of (1+i)...
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#differentialequation #maths #math #blackpenredpen #mathforfun
#complexnumbers #imaginarynumber #calculus #hardmath
Forgot to mention, y=0 isn’t a missing solution since 0^i isn’t defined. See here ruclips.net/video/-ExXldVjYp8/видео.html
Please do this integration of 1/(2sinx+cosx+3)
@@vortex370 you can maybe substitute sin(x) with "u" and rewrite the cos(x) term with "(1-u^2)^0.5". Then replace dx with du with the Jacobian and rewrite the integral as much as possible until you might be able to perform a partial fraction decomposition. Keep in mind that cos(x) is then du/dx which might be very useful here. This is how I would solve this but it surely becomes complicated after a few steps :(
@@DanielMaurerDiabolo can u solve it for me
@@DanielMaurerDiabolo as it easy to say but hard to do
@@DanielMaurerDiabolo and I don't think it's a question which can be done with partial fraction
Step 1: Isolate y
Step 2: Get answer
Step 3: Simplify
Step 4: Realize you could have stopped at Step 2
I don't think he really simplified it. The "simplified" solution after step 4 seems more complicated than what he had at step 2
Hahah nice!
Step 5. Work on a definition of "simplify"
@@trueriver1950 Step 6: Give Up
@@anshumanagrawal346 Step 7: Contemplate on your life.
dlnr = department of land and natural resources
😂 you should change your subject to geography
dlnr - донецкая и луганская народные республики
"we're all adults now so be sure you use radians"
edit: corrected the quote at 5:40
15 yo🙁
@Milky Way Galaxy I was just quoting his comment at 5:40 because it was funny
@@alessiokrolikowski5501 I was just quoting his comment at 5:40 because it was funny
@@MrCoxmic i know
1:23 “i don’t like to be on the bottom, i like to be on the top” 😏
Julian Westerlund I snorted when I heard that
I don't get it.
drz It's okay to be innocent
Amit Tiwari well yes but also no
@@stza16
"Bottom" and "top" often refer to the positions that people have during sex.
3:30 "This right here has absolutely no numbers" me: "e"
Jeremy Nehring Lol yea I realized that too after i said that
me too: i
Interesting. One question: why isn’t the complex power of a complex number multivalued? Or are you only picking a specific branch?
Ming Miao it is. Yes I only consider the principle branch.
blackpenredpen I love you and your videos
I'm always impressed when he does these long videos without any notes at all!
he knows how to do it before hand
"You do not have to solve this in real life." Obviously.....it's in the imaginary world
Imaginary numbers are not imaginary they are just as real as real numbers
To be a little clearer, "imaginary" is just a name given to such numbers, and has no connection with any numbers existing or not existing in our world. TL;DR - Real and imaginary are just names, with no other meaning.
@@green0563my guy really wrote tldr
Mr blackpenredpen it's corona time we need daily videos I know I'm disturbing you but I like your math questions plz some integrals question 😘😘😘😘
I pray that h e does not take this as an incentive to try to crack corona as a differential equation. works. but damn i´d kill myself if i had to unsubscribe to another channel because it´s corona fucking everywhere
@Lu Ste we'll pray in the name of jesus
"Simplified"
To mathematicians, "simplified" pretty much just means "easier to deal with", no matter how messy it looks.
Omg I had a panic attack when you wrote the formula because I thought the d theta in the exponent of e was a differential
Step 1: isolate y
Step 2: get answer
Step 3: "simplify"
Step 4: realize that you could stop at step 2
Step 5: review the definition of "simplify"
I just want to thank u cuz you always help me to improve my math techniques and i'll never forget your good for me
I had a feeling that pi would pop up in this problem.
I'd love to see a differential equation like this as a bonus question on a test 😂😂😂
I love your videos, and it makes me happy whenever I see just how much you love math. Your videos have shown really cool answers to questions I have had myself, and I love imaginary numbers, so these videos are amazing :)
1:24 well I’ve never heard someone say that but you do you I guess
I adore videos with problems that are unlikely to be undertaken by anyone due to their complexity. It's just more fun watching complex, rather than real problems, if you know what I mean.
My favourite Calculus Teacher is backk😍🔥
Whenever he says that he has done something in another video, I already know beforehand which one :P
Continue the great work! Cheers and #stayhome!
I adore this "pointless" videos of yours, man.
"i don't like to be on the bottom, i like to be on the top"
make your viewers blush
A convention I rarely see is replacing any expression of the form cos x + i sin x with the form cis x. Especially useful when x itself is “complex” expression as in the example illustrated here.
Is there a way to find out what proportion of all possible rational numbers are in their lowest terms?. Essentially if you have x/y where x and y go from 1 to n, you get n² possible rational numbers, of which only some instances are in their lowest terms. I guess it just means of all possible combinations of x and y where x and y go from 1 to n, how many combinations are co-prime relative to n². And does that have a limit as n approaches infinity?
I would simplify the last bit in the cos and isin to be (ln4 -pi)/8 because to me that is more straight forward.
(2 ln2 - pi)/8 😎
Buen vídeo. Excelente, saludos desde Ecuador.
blackpenredpen, can you make a video for how to add vectors (or complex numbers) while still in polar form without transforming into cartesian form?
Your videos are the best!
7:00 Third try? You actually recorded this twice before (but scrapped them)?
The higher order differentials have a nice pattern y^(2) = i y^(2i-1) and y^(3) = i * (2i-1) y^(3i-2) and if you continue you get a sort of diagonal factorial involved y^(N) = Product (n=2 to N, ((n-1)*(i-1)-1)) * y^(Ni-1) - it looks like a pattern that should involve a combination of cosh sinh, cos and sin and a whole lot of fun
What threw me was the introduction of a blue pen. After that I never really recovered.
You can write the answer as products of cosh, sinh, cos and i*sin since e^(pi/8)=cosh(pi/8)+sinh(pi/8)
I love your videos
I had to pause the video @9:18 and see how to eliminate the 'i' in the exponent means so much terms, the imaginary is really complex, maybe it's because it like a 4D in Math, complex indeed; fascinating...
It's actuallly nice to see BPRP solving ODEs
Enemy Hunter Official/EHO EHOfficial/ thanks!
At 0:49, you should further explain your algebraic notation of multiplying (1-i) by both sides. You are moving too fast.
I’m so happy you’re back! I’m on lockdown, and this helps.
y=c+xsqrt(2)
Edit: forgot to put y=
Thanks for your great work in this hard times
(0.976 - i0.218)*sqrt(x+c)*[cos{0.5 ln(x+c)} + i sin{0.5 ln(x+c)}]
Pls solve for x
dx=asin(y)+bcos(y)+cxdy
a;b;c are constants
I got into Cornell for engineering and I just want to thank you for inspiring me.
I did not like your notation for multiplying both sides of an equation by something on the third line of the video, but it is clear enough. Good stuff my man!
Integrate 0 to 1 (x^2(4-x^4)/(1-x^2)^1/2)
Excellent. Thanks a lot.
What does y^i mean?
3blue1brown has a great video on why e^iπ=-1 (and how e^ix = cos(x) + (i)sin(x)). Basically about how Euler's formula corresponds to a unit circle on the complex plane.
I think it might help answer your question. Sorry, I don't know enough to answer it myself. I'll get the URL and be right back.
ruclips.net/video/v0YEaeIClKY/видео.html Here's the video
Is Cis a formal thing for cos + isin ?
My lecturer use Cis so he doesn't need to write that much
I never heard of that notation. I don't think it is very widely used. I don't know anyone who uses it.
Yes. This notation is used. I found the notation in the book "Pre-College Mathematics".
sichka _ It is formally acceptable, but I would not recommend it, since in the end, the readers will most likely be confused and you will need to explain it anyway, not to mention that e^ix is just equally as simple.
Thanka for the explanations :)
I want to see him take the derivative and show that it works.
Pretty cool!
I saw this on my screen and immediately had a heart attack.
You never fail to confuse the hell out of me but make me feel like I can conquer the world.
That thumbnail tho😂😂
"I don't like to be on the bottom, I like to be on top" AYOOOOO-
You could of redefined c to be simpler earlier ie replace c(1+I) with a new c. It’s just a constant that can be any number.
Integration constant is complex too.
5:39 whenever you say that you make me laugh LOL
good job man you are the best calculus teacher i have ever seen
Watching your videos from 🇧🇷🇧🇷
DJ VALENTE DO CHP lol thanks.
I'm pretty sure that this is useful in quantum mechanics
i dunno. Give an example
Really?
@@blackpenredpen
actually yes
well..... sort of
in my classes in effect of potential field on a boson, it was common to use complex numbers, but usually they were only used for multiplication. But my professor went a step ahead and gave a problem involving a very similar pattern as in the video
he said it was not a practical scenario but still thought it was afun problem to give to us
@@divyanshaggarwal6243 which college are you in? Mai Indian
@@me_hanics It is used in the heisenberg matrix mechanics and the schrödinger equation.
What a simple solution for that complex question. No wait, its the other way around: simple question, complex solution..... Eh, both complex
please integrate (4+(cosx)^2)^1/2
amazing... as usual.
Hey bro..na awesome video again!
But I hv a enquiry , did u make videos about Non homogeneous higher order diffrential equations and particular integrals??
If yes, plz make a playlist for that! I am unable to find those videos😣
Can you simplify the other part using the same equation with the imaginary part inside the brackets equal to zero?
The polar version of complex^complex is even more complex.
Can you make a video on math Olympiad books , please.
Hey Steve, Fantastic Equation!
ill never need to know this in real life but it will definitely be useful in imaginary life
Excellent!
It's a complex numbers day!
chill there, differential eqn was already killing me
Make more number theory videos c:
RUclips recommendation is really bizarre. Good thing I understand most of this.
Do could the deducion of exponentiation of two complexs numbers a +bi that you showed in the vídeo please?
I'm not sure what is more impressive doing it this way or taking the derivative of that mess and getting y^i
What is the need to simplify (1-i)^(1/2+1/2i)
Everyday I watch your videos
Now check the answer by differientiation! ;)
We missed you
I could have never “imagined” such a thing if I had seen it 🧐
Now check the answer by differentiation :)
2:20 i don’t like to be in the exponent
Sir you are from which country???????
I think you are from South Korea , am I right??????
This is out of jee syllabus, but RUclips recommended it to me...
Now your subscibers have increased by | i² |
You would think the answer would be more elegant but nope
Apple pen + pinapple pen = blackpenredpen
can you do videos on frobenius method, bessel equations and legendre transforms?
I promise you will never need this. Challenge accepted
NitronNeutron ok!!
Sir pls tell us how the formula comes pls.. Or send the video link
Your a bad man in the math class my brother... Just wish the crew would just fallow the process
Is the constant of integration c a real or a complex number?
The constant of integration can be any number, real or complex. It is common that when the integral is meant to be real-valued, that you'd expect a real-valued constant of integration as well, otherwise the output would be meaningless.
This is how the integral of 1/x relative to x, is ln(|x|) + C. In a general complex case, it is ln(|z|) + i*(angle(z) + 2*pi*k) + C, where k is any integer and c is any constant, real or complex.
Let k = 0, and let C = -i*pi, and you'll see that ln(|x|) is the integral of 1/x, for both positive and negative numbers. Since you cannot integrate directly across the problem point of x=0, you can have a different constant of integration for negatives as you have for positives. Let C = -i*pi*(2*k + 1), and you cancel out the imaginary part of the natural log of a negative number.
Just by eyeballing the whole (1-i)^1/2+1/2i seems like its about 1. Anyone who botherd to compute that want to let me know?
why not use complex exponential instead of cos + isin ?
How does this jibe with the very different result given by WolframAlpha?
θ=arctan(b/a)
θ=atan2(a, b), to account for all four quadrants.
Hey, i did this
Took ln both sides
Divided by lny so to have i on rhs
Squared both sides to eliminate i and there i was😎
what's the point of expanding out the power? Why not leave the (1-i)^(0.5 +0.5i)
The Double Helix It is not very useful to work with complex exponents if the base is a polynomial.
Do you guys just like watching these or is it just me
This is nice
What if you do idy/idx instead of dy/dx
the i cancels out and becomes dy/dx
dlnr sounds like a railroad line
Do dy/dx = i^y :D