Solving Differential Equations with Power Series: A Simple Example
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- Опубліковано 9 лют 2025
- Here we show how to solve a simple linear differential equation by solving for the Power Series expansion of the solution. This is an extremely powerful approach that can be used to solve nearly any differential equation, even nasty nonlinear equations.
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This video was produced at the University of Washington
%%% CHAPTERS %%%
1:08 Solving Simple ODE with Power Series Expansion
6:40 Recursively Match Coefficients of Each Power t^n
12:50 The Full Solution: An Exponential Function
Have to say out of all lectures I see on UA-cam, your lectures feels like I am right there in front of you when you are teaching... The way you communicate complex topics in simplified way is outstanding!!!!
Absolutely been LOVING your lectures. I'm using them as refreshers, but I wish I could have learned all this brand new from you and these lectures.
Glad you like them!
Best lecturer I've ever had the privilege to listen to - and I've complete Grad school.
Exponential function and its Taylor series are fantastic tools in numerical time integration... what this function does is actually magic...😍😍... Thank you, Steve. 🌹🌹🌺
This is a nice derivation of the taller series.
Thank you for sharing your knowledge! An amazing explanation.
Thank you very much for this lecture...
Homework (Exercise):
In time 5:41:You can kind of convince yourself that the only way for this (xdat(t) = ax(t)) to be true at every instant in time is for these (the polynomials of xdot(t) and ax(t)) to equal each other for every Power of 't'. This is actually a really good exercise to get comfortable with these series and think about if this is true for t equals zero, and one, and two, and three, in all time (you know, 1.1, 1.2, 1.3, for all time), the only way that can be true for all time is that if every single term in these expansions of xdot(t) and ax(t) equal each other.
The casual C3PO made me laugh🤣🤣🤣🤣🤣🤣🤣🤣
Steve, cannot thank you enough!!!
Informative lesson! Thanx! 😊
Amazing. Nice one Steve.
Thanks!
Beautiful. Beautiful, because we need to use an infinity of terms. No human could observe this infinity of polynomials. But, the greatness of the human mind is that it's not nececessary to see them. We just need to conceptualyse, to be logic and coherent. And then by magic, the exponential fonction appears.
Euler was the king of this kind of reasonning. Euler is the guy who transformed calculation in art, who transformed calculation in a new branch of mathematics : The analysis.
absolutely brilliant!
Bravo ! .. and hats off to Euler !
“write down a power series . . . “ Okay, but the one you wrote down I already knew was the power series for e^at.
I can't find the lecture series /playlst about the taylor series on your channel. There's two videos on that topic following this one in the DE playlist... did you mean that? Ok nvm yes it's just a vid and not a playlist. got confused since it came after this lecture.
Nice video! So basically, if we do all this and don’t recognize the series/pattern, we just truncate the series to approximate the solution to the ODE ?
2:05 how valid is this assumption? is there a proof for it?
C0 disappears for derivative of X, why the two side still have same number of terms?
Because the derivative was taken with respect to time (t). In this case c0 was a constant and not dependent on time. The rate of change of a constant is 0, hence why it disappeared.
Also we're assuming it's an infinite series. So not exactly true for you to say equal number of terms.
@@marshall7253 thanks, highest order of X's coefficient must be zero.
Why not using sigma for sums?
C3PO 😆🙌