Complex roots of the characteristic equations 2 | Second order differential equations | Khan Academy
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- Опубликовано: 29 сен 2024
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What happens when the characteristic equation has complex roots?
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Differential Equations on Khan Academy: Differential equations, separable equations, exact equations, integrating factors, homogeneous equations.
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thank you a million times, Khan academy has helped me out through highschool and mechanical engineering school!
please never stop
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Yeah man it is really helpful in learning Mechanical Engineering math
Thanks a lot!!
oh nice I didn't realize that until I read your comment I kinda just took it for granted, thx :)
you rock the math man XD thank you
so complex number minus complex number yields a constant? which is C4 hmm
great
helpful
u are the best
At first I wasn't sure how (C1+C2) and (C1i+C2i) would both work out to be real. Then I realized (C1+C2) and (C1i+C2i) can both be real if C1 and C2 are both complex numbers with equal real components and equally opposite imaginary components (C1=A+Bi and C2=A-Bi).
C1+C2 = (A+Bi)+(A-Bi) = 2A (real)
C1i-C21 = (Ai+Bi^2)-(Ai-Bi^2) = (Ai+B(-1))-(Ai-B(-1)) = (Ai-B)-(Ai+B) = -2B (real)
A and B could be any real constant.
Now the part @ 4:49 makes sense to me. This stuff is pretty fascinating.
Thank you for this clarification!
but what if both c1 and c2 were constants then c1i+c2i would be complex number in the form of (c1+c2)i.....
hero.
Yes, that makes perfect sense since C1+C2 depends solely on A and C1i-C2i depends solely on B, so this substitution does not affect C1+C2 and C1i-C2i still being arbitraty.
yeah this was my exact contradiction coming to this comments section I was like how can they both be real, thanks for answering it already
but what if both c1 and c2 were constants then c1i-c2i would be complex number in the form of (c1+c2)i.....so putting the sconstant c4 will omit the imaginery part of the solution....how is that ?
if anyone sees this, can't we simplify C3cos(ux) + C4sin(ux) into C5sin(ux + C6)??
Explaining this took my teacher about an hour and at the end no one (including myself) got it. This 10 min explanation I got it right away. Thank you, Sal.
I don't know who you are sir! But you have a very very special place in heaven! You just taught me this in half an hour what my teacher couldn't explain in an entire semester! I hope you get everything you need in the world!!!
Euler is pronounced "oiler". I learned more in 3 days of your videos than an entire semester of Diff Eq! Thank you!!
facts
C'mon Sal... Euler is pronounced "Oiler" not "YOO-ler". :)
in the end e^(-x/2) should multiply the whole quantity, right ? I think he might have missed a parenthesis.
Why don't we chose beta to be -sgrt of 3/2 but positive.. because the sign for beta in the r equation was plus or minus
Thank you Khan academy and especially to the brother who explained parts 1 and 2 of Complex roots of the characteristic equations.
@bob9630 It doesn't. Fixing C3 doesn't restrict your possible choices for C4. Take any pair of C3,C4 and you'll get a particular solution of the differential equation.
To state it differently, for any complex numbers C3, C4 you can find complex number C1, C2 such that C3=C1+C2 and C4=i(C1-C2)
pleaaaaaaase answer me,,, is this the same as the d.operator method???,,, and what is the variation of parameters method ??????/ pleaaaaaaaaase people!!!!
if u read this put a thumbs up for a good comment, and khan academy rocks!!!!!! :)
from dubai
Do you have the files of your notes anywhere for the public?
Really, who comes up with this stuff? WHO FIRST THOUGHT TO DO THIS???
I still didnt get it : how C1i-C2i =C4 ??
Thank you very much !! :)
This is barely even the prequel to the movement SAL... having this quality of material so easily accessible to the entire world instantaneously... you my friend have started a revolution. and that word is TOO OFTEN used loosely! lectures will be streamable soon for all universities.. and i can imagine a wall-e type scenario where kids just sit in pods from the moment they can respond to visual and auditory stimuli in the not-so distant future.. and some of u fools r critcizing pronunciation..LOL
i thought Euler was pronounced "OY-ler"
u can say that
Okaaay, I've already understood how Euler is pronounced. You can stop saying this now...
He pronounces it wrong though ^^
The beginning of the name is the same sound you make at the end of the word boy "oy" oyler ie.
when should we solve for 'r' and when should we leave it in complex roots equation.. ?? i mean at 08:30, we could have solve 'r', but instead we write in in complex equation to sub into other equation..
You are beyond AWESOME. Thank you. ^_^
I need simplification for (d^2y/dx^2)-m^2y=0
thanks
This is the best explanation that I have ever seen on this problem, may it be in a textbook or lecture or whatever. Sal, you rock!
AMAZING !
i love you
Class
dude... i love you...im not scared to admit it... true love right here
i have a major exam in 5 days and I am pretty much revising by looking at all your vids. It's helping so much!
Thank you sooooo much for these great vids.
It's worth saying that you could do most of this using one trig identity
Forgot to close a bracket at the end there. Great stuff though!
You are awesome, i completely understood the material. Thank you :)
It can be, it can also be pronounced the way he does.
Quick question about the relationship between C3 and C4. Since C3=C1+C2 and C4=i(C1-C2), are the choices for C3 and C4 truly arbitrary, or does choosing one, affect the choice for the other?
hello.
This is awesome, thanks...
What is the importance of complex roots?
@Rizmaestro ill be doing the same thing :D
THANK YOU!!!!!!
hahaha, like your display name
nice video, thank you sal.
Wouldn’t it be 3i/2 instead of square root of 3/2 times i?
no
Nicely done
You are AMAZING!!!!!
it is
240p :=