I made a mistake at around 49:16 when I wrote the formula for t*. The dimensional version of the correction term with the 4/3 in it should have a (v_0)^3 in the numerator, not (v_0)^2. Sorry!
i.e. the first example showed a function which is analytic and it had this nice uniform property, the essential singularity function did not enjoy this uniform property, so what happens if a singularity is removable?
y’(x) = y(x)^4 + a(x)*y(x)^2 + c(x); y(0)=0 Obviously if y(x)^4 -> ε*y(x)^4 then you have the Ricatti Equation plus some nonlinear perturbation (the quartic term) which can be approximately solved by these methods. But… How do you use these methods to solve the original problem without the epsilon?
Lecture 10 is coming. I prerecorded lecture 11 and did not mean to post it yet. I'm taking it down to avoid confusion and will put it back up later, after lecture 10 is posted.
I made a mistake at around 49:16 when I wrote the formula for t*. The dimensional version of the correction term with the 4/3 in it should have a (v_0)^3 in the numerator, not (v_0)^2. Sorry!
59:15 You can also check out Theorem 9.2 of Verhulst's "Nonlinear Differential Equations and Dynamical Systems" for convergence. :-)
@1:02:50 , say x =2 , then e^2 / ( 1 - e^2 ) , wont always be > 1 ? Am I missing something here?
When we solve odes and pdes in perturbation theory, do we always assume that each function is of order 1
is the non uniformality of the perturbative series in the end a consequence of f having an essential singularity for \epsilon
ightarrow 0?
i.e. the first example showed a function which is analytic and it had this nice uniform property, the essential singularity function did not enjoy this uniform property, so what happens if a singularity is removable?
skip to 24:10 for actual method of using perturbation theory.
y’(x) = y(x)^4 + a(x)*y(x)^2 + c(x); y(0)=0
Obviously if y(x)^4 -> ε*y(x)^4 then you have the Ricatti Equation plus some nonlinear perturbation (the quartic term) which can be approximately solved by these methods. But… How do you use these methods to solve the original problem without the epsilon?
Math aside (which is great), the physics in this video is superb.
Thank you !
On the top left it says Sun 14 Mar
I was born on Sunday 14 March of 1999
Maybe shouldn’t gloss over use of negative binomial theorem…
Is this lecture 10 or that one is missing?
This is Lecture 11. Lecture 10 has now been posted.
@@stevenstrogatz1 Thanks!
Say, was there a lecture 10? Or is this misnumbered and supposed to be lecture 10?
Lecture 10 is coming. I prerecorded lecture 11 and did not mean to post it yet. I'm taking it down to avoid confusion and will put it back up later, after lecture 10 is posted.