User:Egm6322.s12.team2.steele.m2/Mtg13

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

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Note: Number [[media:Pea1.f11.mtg12.djvu|Eqn(1) p.12-4:]]is a particular L1-ODE-VC, more general than [[media:Pea1.f11.mtg11.djvu|Eqn(3) p.11-3:]], which is similar to the L1-ODE-VC in King 2003 p. 512, since to obtain (3) p.11-3, we need to require that $$\bar b(x) \ne 0, \ \forall x$$ Q: Is it possible to exhibit a more general class of N1-ODEs that are either exact, or can be made exact by the IFM? (See p.12-5) IFM: Due to Euler, who made important contributions to many areas of mechanics (and applied mathematics), in particular celestial mechanics. “When a first order equation is not exact it is often possible to multiply the equation by a quantity, called an integrating factor, that makes it exact. Though integrating factors had been used in special problems of first order ordinary differential equations, it was Euler who realized (in the 1734/35 paper) that this concept furnishes a method … Clairaut independently introduced the idea of an integrating factor in his 1739 paper … All the elementary methods of solving first order equations were known by 1740.” Kline 1972, Mathematical thoughts …, v.2, p.476

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Of Euler, Clairaut, and the Moon… “In a dispute with more than just scientific import, Alexis Clairaut, Leonhard Euler, and Jean le Rond d’ Alembert each employed their own strategies to establish that they were the first to understand a puzzling feature of the Moon’s orbit.” Bodenmann, The 18th-century battle over lunar motion, Physics Today, Jan 2010. Back to creating a more general class of N1-ODEs that are either exact, or can be made exact by the IFM. Consider a class of N1-ODEs of the form:

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Where a(x), b(x), c(y) are arbitrary functions. R3.1 Show that the N1-ODE [[media:Pea1.f11.mtg13.djvu|Eqn(1) p.13-2]] satisfies the condition [[media:Pea1.f11.mtg11.djvu|Eqn(2) p.11-2:]] that an integrating factor h(x) can be found to render it exact, only if $$k_1(y) = d_1$$ (constant). Show that (1) p.13-2 includes (1) p.12-4 as particular case.

Integration constants

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R*3.2: Show that (1) either is exact, or can be made exact by the IFM. Find the integrating factor h. R*3.3:

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1.	Find an N1-ODE of the form (1) p.13-2 that is either exact or can be made exact by IFM. (See R3.1) 2.	Find the first integral $$\phi(x,y) = k$$ R3.4: Construct a class of N1-ODEs, which is the counterpart of (1) p.13-2, and satisfies the condition (1) p.11-3 that an integrating factor h(y) can be found to render it exact.

=Derivation of a more general class of N1-ODEs that are either exact, or can be made exact by the IFM= Based on the condition [[media:Pea1.f11.mtg11.djvu|Eqn(2) p.11-2]], consider:

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If $$k_1(y) = d_1$$ (constant)  (see R3.1), then (2) p.11-2 is satisfied:

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Hence a more general class of N1-ODEs that are either exact, or can be made exact by the IFM

Where  are arbitrary functions.

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=L2-ODE-VC with missing dependent variable= (1) (2)	P.7-3: p(x) := y’(x)