Higher Order Constant Coefficient Differential Equations: y'''+y'=0 and y''''-3y'''+3y''-y'=0
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- Опубликовано: 2 июл 2024
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In the previous videos in our ODE playlist we have seen how to solve constant coefficient homogeneous second order differential equations like y''+2y'+y=0. We saw how there were three options, distinct real roots, repeated real roots, and complex pairs to the characteristic aka auxiliary equation. The same basic idea applies for higher order differential equations as well, its just that we have more roots but they are all the same possibilities.
0:00 Intro
1:00 Example 1
6:40 Example 2
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Thanks for doing these videos. I am a self-taught roboticist. You have become my go-to channel for math. Whenever I come across something in my reading I don't understand, I go to your playlists and look for relevant topics. It is pretty common for me to watch a video a couple of times over a few days before I become comfortable relating the material to my work. If I am still struggling, I look for an OCW lecture. With the good foundation your material provides, even the stuff from MIT is reasonably understandable.
Very cool, thanks for sharing!
@@DrTrefor Quick follow-up to my earlier comment. I like your use of graphics. I occasionally print screenshots. Then I use the screenshot as mnemonic devices to recall an aspect of a particular topic.
I put the screenshots in a pile near where I eat breakfast. Every morning as I eat breakfast I go through the pile and try to answer two questions. Why did I take this particular screenshot and why is it important to what I am trying to do?
Viewing these vids after some exposure to ODEs has been great. It's really nice to already have some context as it makes the inform stick a bit better the second (or third) time round.
A small mistake at 5:17 , the constants should be c1, c2, and c3.
Thanks so much for all of your playlists! They're really helpful for my self-studying!
also, at the same spot, the formula is e^αt*cos(βt). He forgot the t term in the exponential
@@perseusgeorgiadis7821 AHH,... That's why it felt weird.
TY! I was so confused when I saw that. GPT explained it to me haha
These videos are neat, tidy, to the point, and clear. That's what we want! 👍
I've already taken Differential Equations but I love how you break down your explanations, so I watch for fun. Thanks for another great video!
Great to hear!
same.
If you were an instructor of my diff class, even if the class was at 8.am all of your coffes'd be from me. Thank you so much man i really appreciate your work and efforts.
I'm very new to differential equations and I swear your videos are by far the best explanations I've seen so far, thanks a lot!
Very Nice and crisp explanation of how the basic idea of solving homogenous homogeous ODEs (with constant coeff) of orders>2 is SIMPLY an extension of finding the solution of ODE's of 2nd order.
I recall in college we learnt about the 'D' (differential) operator in the similar to the "r's" in "characteristic" equation.
I would request you to make a separate video on this 'D' operator (attributed to the genius American engineer/mathematician Oliver Heaviside), and its inverse '1/D' ( which is the integral operator) and use the Laplace Trasform and makes solving homogenous ODE's of any order look so simple- almost like school algebra and is a standard tool used to solve complex circuits in electrical engineering.
Looking forward to more gems from you Prof Bazett
👍
I liked this you have helped me understand these ODEs
i loveu fr u saving me academically
Is it supposed to be C3 and not C2 at 5:12?
Nice catch! Yes indeed:)
Beautiful! Thank you!
Professor Bazett, thank you for an excellent video/lecture on Higher Order Constant Coefficient Differential Equations. DR. Bazett, when you factored r^3 +r equal to zero, you have r(r+i)(r -1)equal to zero) instead of r(r + i)(r - i) equal to zero. There is also two c2 in your final solution. Please correct these small errors in the video.
Thanks a lot sir 🔥🔥🔥
Nicely done, sir. Keep up your good work
Thanks, will do!
A mighty beard and a mighty video. Brilliant!
haha thanks!:D
Didn't expect to find better than khan academy
At 5.18, Should the third constant in the final answer be C3, instead of being equal to the second constant C2 ?
at the end, when finding the c4 term, instead of multiplying the c3 term by t, could you multiply the c1 term by t to get c4*t as your final term instead?
Respect = Dr. Trefor Bazett (with love )
Nice video! Beyond pure mathematics, just curious what physical phenomenons can be modeled with higher ordered (>2) differential equations?
All sorts of things in fluid dynamics in particular. Stopping at acceleration is fairly arbitrary, we can also ask questions like how does acceleration change in time?
e to the it power is cos(t) + isin(t). you're supposed to add an i in front of the sin(t)
Thank you very much!
You're welcome!
in example 1 could you write c_3 instead of c_2 in the awnser y=c_1 + c_2*cos(t) + c_2*sin(t) ...or what happens to c_3?
That is typo. You are right.
What pen you using i also want to buy to teach my students
How do you know you can divide by a complex number is that still a linear combination
A complex number is still just a constant. Complex coefficients are still valid coefficients to use for constructing linear combinations.
made quite a simple mistake at 5:17, it can happen. but please add a disclaimer next time while editing as it can be very confusing.
4:58 should be e^(αt) instead of e^α
if you can always multiply the solutionr by t doesn't it mean that you have an infinite amount of solutions?
There is but they are all just linear combinations of the two linearly independent solutions
Good question, but the answer is no. Multiplying by unnecessary factors of t, will produce "solutions" that don't work.
As an example, consider:
y" + 6*y' + 8*y = e^(-2*t)
The correct solution is:
y(t) = A*e^(-4 t) + B*e^(-2*t) + 1/2*t*e^(-2*t)
Since our given RHS matches one of the homogeneous solutions, we have to multiply by t to create a linearly independent term to account for the overlap.
What if we tried C*t^2*e^(-2*t) as one of the solutions?
d/dt C*t^2*e^(-2*t) = C*(2*t - 2*t^2)*e^(-2*t)
d^2/dt^2 C*t^2*e^(-2*t) = 2*C*(2*t^2 - 4*t + 1)*e^(-2*t)
See if this works, as a solution to this diffEQ:
2*C*(2*t^2 - 4*t + 1)*e^(-2*t) + 6*C*[(2*t - 2*t^2)*e^(-2*t)] + 8*C*t^2*e^(-2*t) =?= e^(-2*t)
Cancel the common factor:
2*C*(2*t^2 - 4*t + 1) + 6*C*[(2*t - 2*t^2)] + 8*C*t^2 =?= 1
Expand, and combine like terms:
4*C*t^2 - 8*C*t + 2 - 12*C*t^2 + 12*C*t + 8*C*t^2 =?= 1
2*C + 4*C*t =?= 1
We have a linear expression of t on the left, and just a constant on the right. No matter what we make the unknown C equal to, we can't make a multiple of t^2*e^(-2*t) be a solution to the original DiffEQ.
Doing this by assuming a solution of C*t*e^(-2*t), we can find a value of C that works.
Combining it's derivatives per the LHS, we get: 2*C*e^(-2*t), which is proportional to the given RHS. We have no issue letting C=1/2, to find the particular solution, by having guessed C*t*e^(-2*t).