CORRECTION: At 9:15 I briefly posted the solution to the undetermined coefficients example - but I posted the solution for the wrong example! The right solution can be found at 4:43 of this video: ruclips.net/video/AgyeJEO2a-k/видео.htmlsi=lc1wKQW5EFzeoJws&t=283
that's kinda why I made this video. Sure I got a lotta more normal lectures that go deep on one method, but you kinda got to see them all beside each other to see the larger pattern
I used "Elementary Differential Equations and Boundary Value Problems" (3rd ed.) by Boyce & DiPrima in a DE course offered by the Math Department; and "Mathematical Methods in the Physical Sciences" (2nd ed.) by Boas for the Mathematical Methods course offered by the Physics Department. They both have very applied approaches to the subject. "Partial Differential Equations in Physics" by Arnold Sommerfeld is based on his lectures on Theoretical Physics. It, like other volumes in the series, illustrates the thought-processes of a truly great mind; he was one of the strong bridges between Classical and Quantum Mechanics.
I doubt I will need to solve differential equations ever again but I am almost nostalgic for what I used to be able to do. That final checklist - that's so powerful it ought to be banned (or memorised).
Dear Professor Bazzet, I am such a fan of your contents. I've watched all of your series of courses including Differential Equations, Multivariable Calculus, Linear Algebra, and some of the Discrete Maths courses since I was a freshman in college. You are the reason why I could found interest in Math and your courses helped me so much in understanding these concepts and I basically watched your videos only for my exams because of comprehensiveness and it's really fun and easy to understand. Now as I am about to go on to pursue some higher level courses like graph and number theory, stochastic processes and so on, I'm sad to see that you might not be able to accompany me on my journey with some of these higher level courses. But I am really so thankful about everything you've done and hopefully I can survive my future math courses with your inspiration, thank you so much again. Greatest Inspo!!!!!!!!
This is of great help of math students!! Thank you very much! It will be very helpful if you make similar videos on other topics and cover it briefly with the most important material. Very appreciated!
Where were you when I took ODEs? That class was so hard for me, and this would have been huge. Honestly, the other upper division math class that killed me was probability theory and mathematical statistics. I would love to see you do a course on that, just like your ODEs course here. I dont feel like any other youtuber has done much there either. I do know way fewer people ever take this course, but man it was hard. I remember one test, i studied and studied and studied for... and i couldnt even start the first problem. That hurt.
@DrTrefor man, please do some probability theory. Really the math stats stuff. The counting and conditional probability stuff was never an issue. It was the math stats stuff. And I graduated with a pure math degree. That class really hurt me. Thanks Dr Trefor. Love what you do.
Great job Prof. Just like ode, Laplace, Fourier series can you start a new series on PDEs? I want to learn PDEs from you in depth before i die. I can die in peace once i know it in enough detail.
I didn't understand why the case of a delta-like inhomogeneity term, call it p, can't be treated with variation of parameters (VoP) until I worked out an example. OTOH, the case where p is discontinuous can be treated with VoP , just solve over the open intervals where p is continuous. Then use the fact that y^(i) (i ranges from 0 to n-1) are continuous everywhere even at the points of discontinuity of p to find the -general- full solution. This method only works when p is piecewise continuous. Good luck trying to solve an equation with p = Dirichlet's function. 💀
All of them are linear, separable or just Bernoulli. I have at least 400 tasks for nonlinear ones that contains a different approaches, I think it will be much more interesting
Oops - I was copying and pasting from an old video and completely copied over the wrong part of the problem! Yes just ignore that whole slide entirely:/
No general method. Looks like sin(x) is one of the solutions, the other is probably quite messy unless there is a clever solution that isn't immediately obvious. Series solutions can work if you only need a few terms, or numerical methods of course.
I am sure I have seen a video on Math 505's channel about a method to solve 2nd order ODEs with variable coefficients, I don't remember the method but I can find the video .. *Going to another tab and searching for the video* Can you believe it!! He treated the exact same example!! Title: Abel's brilliant trick for solving differential equations
Long ago in 1985, we used Boyce and Diprima's textbook. I went home for Spring Break and solved every problem in the textbook over the holiday. Practice was all that was needed to finish very quickly on the exam. But now, most of the methods I do not use much nowadays except for the y = Sum a_n*x**n method. Instead I use sympy.dsolve or mostly just Euler's method because one can solve dy/dt = f(t,y) which describes most of the diff equations I see. I read Lee's Smooth Manifolds (first Topological Manifolds then Differential Geometry) textbook as a fun introduction to symplectic manifolds which naturally led to lots of usage of diff eqs. Is ODEs the most complicated topic in mathematics? Algebra seems to reduce down to CompSci and Abstract Data Types or High school Algebra. Analysis reduced to Calculus. Geometry reduced to Algebra. Commutative Algebra to Algebraic Geometry to Groebner Bases. I guess I am saying in this comment that I like applications and it seems to be easier to do Euler Method (or RK4) than the long algebraic technique by hand, pen, paper, and robotics writing commands much like PDF or LaTeX. Is Mathematics just writing on paper so hence is it just Mechanical Engineering, perhaps with Artificial Intelligence used to control the Mechanics?
@@DrTrefor I really like numerical methods. I have been using Microsoft Copilot (a large language model) and some times I ask it to "write a python3 script to ..." sometimes asking it to build a system of ODEs to model something. At the moment, now I am not reading Mathematics but instead a Neuroscience textbook, but to mix the two topics together, there is Gabbiani and Cox's Mathematics for Neuroscientists which is a nice Applied Mathematics. It is not exactly Logan's Applied Mathematics, but a specific application. But I guess I am exploring intelligence and hence Neuroscience and Biology. Hopefully, one just builds a neural network (somehow spiking or artificial neural network) and output is something intelligent. I have not learned exactly how the large language models work however. To me, they use something like a Markov Property and conditional probabilities. But for ODEs, Biology is great application---Britton's Mathematical Biology and Klipp's Systems Biology to name a couple applications.
CORRECTION: At 9:15 I briefly posted the solution to the undetermined coefficients example - but I posted the solution for the wrong example! The right solution can be found at 4:43 of this video: ruclips.net/video/AgyeJEO2a-k/видео.htmlsi=lc1wKQW5EFzeoJws&t=283
Right before my ODE final? Thank you Dr. Bazett!
I saw the spike in my ODE playlist and was like I have to release this today:D
these 13 minutes made more sense than my entire semester of lectures, despite being less in depth
that's kinda why I made this video. Sure I got a lotta more normal lectures that go deep on one method, but you kinda got to see them all beside each other to see the larger pattern
@@DrTrefor for sure! when you're in the thick of things, it's easy to feel lost. Kind of a missing the forest for the trees situation
I have my ODEs final on Monday, and your happen to drop this today! Thank you so much Dr. Bazett, you are my hero.
Amazing - good luck!
I used "Elementary Differential Equations and Boundary Value Problems" (3rd ed.) by Boyce & DiPrima in a DE course offered by the Math Department; and "Mathematical Methods in the Physical Sciences" (2nd ed.) by Boas for the Mathematical Methods course offered by the Physics Department. They both have very applied approaches to the subject.
"Partial Differential Equations in Physics" by Arnold Sommerfeld is based on his lectures on Theoretical Physics. It, like other volumes in the series, illustrates the thought-processes of a truly great mind; he was one of the strong bridges between Classical and Quantum Mechanics.
Boyce DiPrima is definitely still in use widely!
I have to credit you mate for being strategic with your upload timing. Very Clever! 😉
I doubt I will need to solve differential equations ever again but I am almost nostalgic for what I used to be able to do.
That final checklist - that's so powerful it ought to be banned (or memorised).
Great synopsis. A one page checklist that points you in the direction you need to go.
I have my final exam on differential equations coming next week and videos like this are a life saver, thanks!!
Dear Professor Bazzet, I am such a fan of your contents. I've watched all of your series of courses including Differential Equations, Multivariable Calculus, Linear Algebra, and some of the Discrete Maths courses since I was a freshman in college. You are the reason why I could found interest in Math and your courses helped me so much in understanding these concepts and I basically watched your videos only for my exams because of comprehensiveness and it's really fun and easy to understand. Now as I am about to go on to pursue some higher level courses like graph and number theory, stochastic processes and so on, I'm sad to see that you might not be able to accompany me on my journey with some of these higher level courses. But I am really so thankful about everything you've done and hopefully I can survive my future math courses with your inspiration, thank you so much again. Greatest Inspo!!!!!!!!
You’re the best Prof ❤
I used to teach ODEs but nearly forgot it all now 😢
got my ode final on tuesday, video helped a lot! thank you!
Great video, Dr. Bazett. Comprehensive recap of everything you need to know about differential equations.
great idea for a video. I took ODE many years ago and this a great review of the basic types. Very helpful!!
My final is on Monday, perfect timing! Thank you Professor!
I'm not even in full time education but I'm still going to watch this video because who doesn't like some good maths?
Ahh this is nice. 2nd order been my favorite so far and I don’t know why
This is of great help of math students!! Thank you very much! It will be very helpful if you make similar videos on other topics and cover it briefly with the most important material. Very appreciated!
Perfect timing! My final is tomorrow.
Oh wow, good luck!
Perfect timing for my final!
perfect timing!
You Made this video at the perfect time
Final tomorrow. Thank you GOAT
Thank you.
Great content.
Please continue to teach us. 👍
I will!
Where were you when I took ODEs? That class was so hard for me, and this would have been huge.
Honestly, the other upper division math class that killed me was probability theory and mathematical statistics. I would love to see you do a course on that, just like your ODEs course here. I dont feel like any other youtuber has done much there either. I do know way fewer people ever take this course, but man it was hard.
I remember one test, i studied and studied and studied for... and i couldnt even start the first problem. That hurt.
I am hoping to do more topics from 3rd/4th year math courses through 2025 for sure
@DrTrefor man, please do some probability theory. Really the math stats stuff. The counting and conditional probability stuff was never an issue. It was the math stats stuff. And I graduated with a pure math degree. That class really hurt me.
Thanks Dr Trefor. Love what you do.
I have mathematical physics final at tuesday, thanks a lot. It'll be perfect for review.
godesend video during finals week ty bro
You dropped this just in time for my diff equ final 😂
just in time; awesome!
Amazing timing
Sir a video of "solving pde methods"
would also be beneficial for students, in case you have time please cover that too
That's coming - but not until jan or feb:(
Thank you I'm always grateful to you sir😊
Great video! Now to do PDEs
I will! But uh not until Jan:D
@ worth the wait! One of my fav courses
YOU ARE THE GOAT
My final is tomorrow!
oh wow good luck!
Great job Prof. Just like ode, Laplace, Fourier series can you start a new series on PDEs? I want to learn PDEs from you in depth before i die. I can die in peace once i know it in enough detail.
This is the plan for Jan-Feb actually!
@@DrTreforomg thank you so much. I will get pde lectures in a month. Wooowwwwwwwww!!!
My final is in 2 days, thank you
This is the best video!!
i owe you my life
i took calculus 3 and differential equations in one semester lmao.. it was freshman year too
I didn't understand why the case of a delta-like inhomogeneity term, call it p, can't be treated with variation of parameters (VoP) until I worked out an example.
OTOH, the case where p is discontinuous can be treated with VoP , just solve over the open intervals where p is continuous. Then use the fact that y^(i) (i ranges from 0 to n-1) are continuous everywhere even at the points of discontinuity of p to find the -general- full solution.
This method only works when p is piecewise continuous. Good luck trying to solve an equation with p = Dirichlet's function. 💀
All of them are linear, separable or just Bernoulli. I have at least 400 tasks for nonlinear ones that contains a different approaches, I think it will be much more interesting
These are meant as the standard topics from an intro ODE course. If you have tonnes of non-linear ODEs, probably that means numerical methods.
@9:21 didn't make any sense to me. How did you go from Ae^2t to A t e^t ?
Yea I got A equalling -1
Oops - I was copying and pasting from an old video and completely copied over the wrong part of the problem! Yes just ignore that whole slide entirely:/
Holy Shi my final is literally tomorrow
Do you have course on linear algebra?
Wish you had stayed at UofT to teach
Miss UofT for sure!
dude my diffeq final was wednesday this would have been nice 😭
lol oh no! hope it went well
@ me too haha. Your Laplace videos were a great help
Too late, I already had to drop the class😢
Need help solving y'' - tan(x)y' + 2y = 0
No general method. Looks like sin(x) is one of the solutions, the other is probably quite messy unless there is a clever solution that isn't immediately obvious. Series solutions can work if you only need a few terms, or numerical methods of course.
I am sure I have seen a video on Math 505's channel about a method to solve 2nd order ODEs with variable coefficients, I don't remember the method but I can find the video ..
*Going to another tab and searching for the video*
Can you believe it!! He treated the exact same example!!
Title: Abel's brilliant trick for solving differential equations
Someone presented another method. If you are interested, check randomzhjioewmx's comment
@@RUclips_username_not_found wtf man thank you so much
@@aaaa8130 what do you think WTF mean?
Long ago in 1985, we used Boyce and Diprima's textbook. I went home for Spring Break and solved every problem in the textbook over the holiday. Practice was all that was needed to finish very quickly on the exam. But now, most of the methods I do not use much nowadays except for the y = Sum a_n*x**n method. Instead I use sympy.dsolve or mostly just Euler's method because one can solve dy/dt = f(t,y) which describes most of the diff equations I see. I read Lee's Smooth Manifolds (first Topological Manifolds then Differential Geometry) textbook as a fun introduction to symplectic manifolds which naturally led to lots of usage of diff eqs. Is ODEs the most complicated topic in mathematics? Algebra seems to reduce down to CompSci and Abstract Data Types or High school Algebra. Analysis reduced to Calculus. Geometry reduced to Algebra. Commutative Algebra to Algebraic Geometry to Groebner Bases. I guess I am saying in this comment that I like applications and it seems to be easier to do Euler Method (or RK4) than the long algebraic technique by hand, pen, paper, and robotics writing commands much like PDF or LaTeX. Is Mathematics just writing on paper so hence is it just Mechanical Engineering, perhaps with Artificial Intelligence used to control the Mechanics?
I’m definitely a fan of numerical methods in many applications. Boyce DiPrima is definitely still in use widely!
@@DrTrefor I really like numerical methods. I have been using Microsoft Copilot (a large language model) and some times I ask it to "write a python3 script to ..." sometimes asking it to build a system of ODEs to model something. At the moment, now I am not reading Mathematics but instead a Neuroscience textbook, but to mix the two topics together, there is Gabbiani and Cox's Mathematics for Neuroscientists which is a nice Applied Mathematics. It is not exactly Logan's Applied Mathematics, but a specific application. But I guess I am exploring intelligence and hence Neuroscience and Biology. Hopefully, one just builds a neural network (somehow spiking or artificial neural network) and output is something intelligent. I have not learned exactly how the large language models work however. To me, they use something like a Markov Property and conditional probabilities. But for ODEs, Biology is great application---Britton's Mathematical Biology and Klipp's Systems Biology to name a couple applications.