The Theory of 2nd Order ODEs // Existence & Uniqueness, Superposition, & Linear Independence
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- เผยแพร่เมื่อ 4 ก.ค. 2024
- MY DIFFERENTIAL EQUATIONS PLAYLIST: ► • Ordinary Differential ...
Open Source (i.e free) ODE Textbook: ►web.uvic.ca/~tbazett/diffyqs
Previously in our ODE playlist, we've studied 1st order differential equations. Now we move to second order differential equations, and specifically focus at the beginning on the major theory. What does existence and uniqueness look like for 2nd order linear ODEs? We see a remarkable property called superposition, and finally we get the big method which is to find two linearly independent solutions to the ODE which will form, by superposition, the the general solution
0:00 Linear ODEs
1:29 Existence 7 Uniqueness
3:38 Superposition
6:18 Linear Independence
7:32 General Solution
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I’m crying because the timing is crazy, I was just about to start with second order but wanted to chill on TH-cam for a bit and then I saw this *uploaded 7 minutes ago* 😩
haha the TH-cam algorithm knows everything!!
The "last time I checked" get me every time! As always, amazing content professor. Btw, I started out learning single variable calc from you, moved on to multivariable calc, then took linear algebra, followed with discrete math, and starting out in ODE's from your videos as well. Can't thank you enough, professor.
Ahhh, I feel like more people need to watch up until this video, this one is really good. Explains a lot of stuff neatly. Let's learn some strategies for solving 2nd order ODE´s
Dr. Trefor Bazett you are amazing. Your Linear Algebra series was a great help for when I was taking that course and here you are helping me in ODE too. It's kind of exciting seeing concepts from Linear Algebra thrown into this course, I feel like having taken the course prior to ODE gives me a slightly better advantage in understanding what's going on too.
absolutely loving these. And so far you're neck and neck conceptually with my class. LOL
nice, that's cool!
The fact that ordinary linear differential equations become harder to study and solve as the order of the equations increase is very reminiscent of equations in algebra: the higher the degree of the polynomial, the more complicated the equations become.
Quite right!
Best explanation I've seen after trying 3 videos on the topic!
Professor Bazett, thank you for a well explained video/lecture on The Theory of Second Ordinary Differential Equation. These Differential Equations consist of multiple solutions.
This was Eye Opener! THANK YOU
Excellent explanation of 2ode theory! Good refresher.
All of your video comes at the right time. In previous semester i had vector calculus course. And this semester have ODE.🙂🙂
haha awesome timing!
Are you planning on going over the proofs? Like picard iteration and stuff? My class goes over it and its very hard to follow especially since it seems to be mostly geared towards the math mayors while im a physics major.
Thanks, I need to check again the video in general understand mathematics it's a bit harder for me for this reason I need to push harder to try to understand
Thank you! I have just started 2nd order linear ode.
Nice timing!
Wonderful, thank you.
At 2:55, shouldn’t there be b0 and b as intervals too along with a? like x is a but y has two points b0 and b1. So two pairs of coordinates should be there in the theorem? I.e (a,b0) and (a,B1)?
Am I missing something?
Thanks a lot sir 🔥🔥🔥
Question: If you have two linearly independent solutions, they span the solution space, as stated in the video, but is this necessary? Must there necessarily be 2 linearly independent solutions for a second order linear ODE, or else you haven't found them all?
Thank u dr.trefor . I have a question , do the initial values have to be evaluated at the same value of the independent variable f(a) and f'(a)
Or it can be evaluated at two different points : a and b f(a) and f'(b) ?
In concept, you could have "initial" conditions that occur at two separate time values. You could even have an example, where the "initial" conditions you are given, are both of the original function at two separate points in time, and not of the function's derivative. The underlying idea is for a 2nd order diffEQ, you need two pieces of information about either the initial condition, or a condition at another input value, to lock down the two unknown coefficients in the general solution. In a general sense, for an nth order diffEQ, you need n-conditions to lock down the general solution to a specific solution, as there will be n arbitrary constants in the general solution.
It is tradition to give initial conditions at t=0, but they don't necessarily need to be at t=0, and they don't necessarily need to both be at the same time. Initial conditions at t=0 help you with Laplace transform, as conditions at other points in time make Laplace transform problems require an indirect method to solve, while initial conditions by strict definition at t=0, can be solved more directly.
thank you fro your great lessons ...i have a question please : what if the two solutions y1 and y2 were lineary dependent , what would be the general solution? ...thx
You would have to keep searching for a second linearly independent solution
I have a question... Existence and Uniqueness theoram implies there exists only 1 solution of a linear second order differential equation..but then you said there can be two linearly independent solutions... Don't these contradict each other???
If it is nth order, you get n linear independent solutions
Thank you ! My dear Professor 👍👍👍👍👍
You're most welcome!
Very engaging video. Thanks Prof
Glad you liked it!
Peace & love 💕
Hi Dr. Trefor!! Are you planning to make any playlist on real analysis in the future? :))
Yes! I really want to! It isn't on the schedule yet though since i want to do so much lol
4:23 by "step away" do you mean turn the camera off or did you actually like move lol
@ 0:46 x is not the dependent variable
what changes for nonlinear
Why is it that the general solution is spanned by only two linearly independent solutions for the second order DE? What assures us that the span of the two linearly independent solutions spans the entire space of solutions?
I know that for nth order the general solution is written as a linear combination of nth solutions, but how to prove it?
My comment seams like a mess, but I hope someone can help :).
Thank you very much
You are welcome!
OF COURSE, THAT RHS-COEFFICIENT OF 1 = y^0 (as if you had taken f(x) to the LHS).
Hello professor, you mean a second order differential equation can't have 3 independent answers? Why not?
nice video dr. !
Thank you 🙂