User:Egm6322.s12.team2.steele.m2/Mtg34

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg34.djvu|Page 34-0]]
Plan: $$\Rightarrow$$Variaton of parameters: Reduction-of-order method 2 History: p.32-3, p.32-4 Important homogeneous L2-ODE-VC Non-homogeneous L2-ODE-VC Given the 1st homogeneous solution $$u_1(x)$$ Find the solution y(x) Deduce 2nd homogeneous solution $$u_2(x)$$ Deduce particular solution $$y_P(x)$$ $$\Rightarrow$$Discovery of Neptune: A modern, corrected story

[[media:Pea1.f11.mtg34.djvu|Page 34-1]]
Variation of parameters: Reduction-of-order method 2 History: p.32-3, p.32-4 =Important homogeneous L2-ODE-VC:= Legendre, Bessel, Laguerre, Hermite, etc. 	$$y\prime\prime + a_1(x)y\prime + a_0(x)y = 0$$ [[media:Pea1.f11.mtg5.djvu|Eqn(1) p.5-5]] Given the 1st homogeneous solution $$u_1(x)$$ Find the 2nd homogeneous solution $$u_2(x)$$ such that Assume the full homogeneous solution of the form: In which U(x) is an unknown to be found. Instead of doing the homogeneous case in (1) p.5-5 (see F09 Mtgs 17, 18, 19, and King 2003), let’s consider directly the non-homogeneous case, and then recover the homogeneous case as a particular case.

[[media:Pea1.f11.mtg34.djvu|Page 34-2]]
=Non-homogeneous L2-ODE-VC= Given the 1st homogeneous solution $$u_1(x)$$ Find the 2nd homogeneous solution $$u_2(x)$$ Consider the solution of the form: U(x) = unknown to be found

[[media:Pea1.f11.mtg34.djvu|Page 34-3]]
Summing up (3)-(5) p.34-2: $$u_1(x)$$ homogeneous solution, hence (1): Missing dependent variable U(x) (1) and (3): L1-ODE-VC

[[media:Pea1.f11.mtg34.djvu|Page 34-4]]
Solve for Z(x) given by [[media:Pea1.f11.mtg11.djvu|Eqn(1) p.11-5]] with h(x) given by [[media:Pea1.f11.mtg11.djvu|Eqn(3) p.11-4]]. Note (3) p.11-4 and R*2.15 p.12-2:

[[media:Pea1.f11.mtg34.djvu|Page 34-5]]
(1)	P.11-5:

Note (3) p.34-3: (2)	P.34-2: Solution for (1) p.34-2 2nd homogeneous solution: $$u_2(x) := u_1(x) \int \frac{1}{u_{1}^ (x)} \exp \underbrace{\left[ - \int a_1(x) \, dx \right]}_{\displaystyle-\bar a_1(x)} \, dx$$

[[media:Pea1.f11.mtg34.djvu|Page 34-6]]
Cf. King 2003 p. 6 (1.3) for the 2nd homogeneous solution. Particular solution:

Note; With the trial solution (ansatz) (2) p.34-2, we can also refer to the method of variation of parameters as the method of undetermined factor (or even undetermined coefficients, but reserve “coefficients” for the case of constants, instead of functions).

[[media:Pea1.f11.mtg34.djvu|Page 34-7]]
=Discovery of Neptune: Irregularities of Uranus’ Motion= See illustration. http://en.wikipedia.org/wiki/File:Gravitational_perturbatino.svg “… two planets orbiting about a common star. The outer planet takes more time to complete an orbit than the inner planet, so once per orbit the inner planet overtakes the outer planet. When the planets are at A, the outer planet exerts a gravitational perturbation that accelerates the inner planet advancing the body ahead of its normal path. When the planets reach B, the reverse is true and the inner planet is decelerated…”

[[media:Pea1.f11.mtg34.djvu|Page 34-8]]
Discovery of Neptune: A modern, corrected story A pilfered planet; the Brits stole Neptune “… Adams utterly failed to communicate his results forcefully to his colleagues and to the world. … Discovery thus has a public as well as a private side. Adams accomplished only half of this two-part task. Ironically, the very personal qualities that gave Le Verrier the edge in making the discovery – his brashness and abrasiveness, as opposed to Adam’s shyness and naivete’ – worked against him in the postdiscovery spin-doctoring. The British scientific establishment closed ranks behind Adams, whereas Le Verrier was unpopular among his colleagues.

The tale also shows the role of luck in discovery. In a sense, neither Adams nor Le Verrier really predicted the position of Neptune. Both greatly overestimated the planet’s actual distance from the sun and succeeded in getting the longitude almost right only because of a fluke of orbital timing. Such things often happen in science (and indeed occurred in the discovery of Pluto nearly a century later).

[[media:Pea1.f11.mtg34.djvu|Page 34-9]]
Now that the passions stirred up by the international rivalries of the 1840s have died down and the original documents have once more been made available for historians to study, we can affirm that Adams does not deserve equal credit with Le Verrier for the discovery of Neptune. That credit belongs only to the person who succeeded both in predicting the planet’s place and in convincing astronomers to search for it. The achievement was Le Verrier’s alone.” Sheehan et al. 2004, The Case of the Pilfered Planet, Scientific American http://www.scientificamerican.com/article.cfm?id=the-case-of-the-pilfered Discovery of Neptune (Wikipedia) http://en.wikipedia.org/wiki/Discovery_of_Neptune