User:Egm6321.f10.team4.Yoon/Mtg12

 Mtg 12: Thu, 16 Sep 10

[[media: 2010_09_16_14_59_27.djvu | Page 12-1]]
Remark: [[media: 2010_09_16_13_50_21.djvu | Eqn.(6) p.11-3]] is more general than [[media:2010_09_14_15_00_52.djvu | Eqn.(5) p.10-2]] (K.p.512) since the latter needs the condition $$\bar b(x) \neq 0 \ \forall x$$ End Remark

Question: [[media: 2010_09_16_13_50_21.djvu | p.11-4]] continued. Yes

= Class of N1-ODE either exact or can be made exact by integrating factor method =

(Euler, Celestial mechanics)

where $$\displaystyle a(\cdot), b(\cdot), c(\cdot)$$ are arbitrary functions.

Application:

[[media: 2010_09_16_14_59_27.djvu | Page 12-2]]
F09: Show Eqn.(1) is either exact or can be made exact

Derive Eqn.(1) - Eqn.(3) [[media: 2010_09_16_14_59_27.djvu | p.12-1]] consider

[[media: 2010_09_16_14_59_27.djvu | Page 12-3]]
Eqn.(6) [[media: 2010_09_16_14_59_27.djvu | Page 12-2]]:

Use Eqn.(1) and Eqn.(3) in [[media: 2010_09_09_13_54_49.djvu | Eqn.(3) p.8-2]](1st condition of exactness) => [[media: 2010_09_16_14_59_27.djvu | Eqn.(1) - Eqn.(3) p.12-1]]

= L2-ODE-VC with missing dependant variable =

[[media: 2010_09_16_14_59_27.djvu | Page 12-4]]
Let $$\displaystyle p(x) := y'(x)$$ [[media: 2010_09_16_14_59_27.djvu | Eqn.(4) p.12-3]]: $$\displaystyle \underbrace{a_2(x)p'+a_1(x)p = b(x)}_{\color{blue}{\text{L1-ODE-VC}}}$$