The Theory of 2nd Order ODEs // Existence & Uniqueness, Superposition, & Linear Independence

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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[fea]

I’m crying because the timing is crazy, I was just about to start with second order but wanted to chill on YouTube for a bit and then I saw this *uploaded 7 minutes ago* 😩

[leadersheir]

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.

[j.o.]

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

[MrEasybreezey]

absolutely loving these. And so far you're neck and neck conceptually with my class. LOL

[glib]

Best explanation I've seen after trying 3 videos on the topic!

[georgesadler]

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.

[richardneifeld]

Excellent, overview, clear, and concise. Thank you!

[Syndicalism]

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.

[angelmendez-rivera]

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.

[tahmidshahriyar]

All of your video comes at the right time. In previous semester i had vector calculus course. And this semester have ODE.🙂🙂

[synapticmemoryseepage]

Excellent explanation of 2ode theory! Good refresher.

[Alannnn]

Thank you! I have just started 2nd order linear ode.

[andrestherock]

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

[TheMvlproductionsinc]

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.

[dwaipayansharma]

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???

[NUGGet-]

4:23 by "step away" do you mean turn the camera off or did you actually like move lol

[mohfa]

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

[fahrenheit]

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?

[elle]

Hi Dr. Trefor!! Are you planning to make any playlist on real analysis in the future? :))

[fatemekashkouie]

Hello professor, you mean a second order differential equation can't have 3 independent answers? Why not?