Talk:Method of dominant balance

Confusing assumption in example

Hi,

I find the way the example ODE is treated rather confusing. The assumption $S_{0}^{'}\sim (S_{0}^{'})^{2}$ comes out of nowhere. Why not simply drop the terms which go as $1/x$ given that we are investigating the limit $x\rightarrow \infty$? Then, the ODE becomes separable once on changes variables to $T=S^{'}$. Would this not be a better way to illustrate the method?

Please sign your messages with 4 tildes. I agree. Dominant balance means you identify 2 dominant terms in an equation solve this simplified equation of 2 terms, substitute the solution into the full equation, and determine the solution's error. This determines the first term of the solution. Repeat to obtain the second term etc. With each term the error becomes asymptotically smaller. You solve for small x near 0 to get the inner solution. Then solve for large x to get the outer solution. This is called the boundary layer approach. The reasoning in this article is unclear. I will fix this when I have time. TMM53 (talk) 18:18, 16 April 2024 (UTC)