More Ordinary Differential Equations - Easy Way 1 This is a short video to cover the transformation method of solving homogeneous linear ordinary diffferential equations (HLODEs), and a particular example thereof, that will be revisited in a different way, later. Ordinary diffferential equations textbooks teach constant coefficient and Cauchy-Euler ODEs and pretty much leave exact solutions there, generally continuing to series solution techniques, like the method of Frobenius. Unfortunately, series solutions don't generally lend themselves to easy determination of zeros, inflection points, periodicity, etc.. Thus, exact solutions, where available may yield additional information, and thus, may be preferred. Using transformations, and the chain-rule for differentiation, you see how HLODEs may be transformed. Consider the Lane-Emden Differential Equation (As noted in Wikipedia and Mathworld, in astrophysics study of stellar interiors, is Poisson's equation for the gravitational potential of a self-gravitating, spherically symmetric polytropic fluid. ) of index 1. Although it clearly is not a general elementary 2nd order HLODE; transforming as illustrated above, it is a rather simple variation on it. My books go into my subjects of interest in greater depth and detail, so would encourage any who find my videos of interest to get my books to delve further at your liesure. /nVisit my author page on amazon.com to find my books available on Kindle in digital and some in print at the website, here: /n http://www.amazon.com/Claude-Michael-Cassano/e/B008MD6CVS/ref=ntt_athr_dp_pel_1 Visit: http://www.barnesandnoble.com/s/claude-michael-cassano?keyword=claude+michael+ cassano&store=allproducts /nfor my books available on NOOK and some in print at barnes & noble.
Channels: Seminars Physics (General) Mathematics
Tags: differential equations Lane-Emden astrophysics spherically symmetric polytropic fluid math
Uploaded by: cloudmichael ( Send Message ) on 04-12-2012.
Duration: 3m 46s