User:Egm6322.s12.team2.steele.m2/Mtg18

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

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Plan: Application: Generate exact N2-ODEs Method: (reverse engineering) Examples

Notes: $$\Rightarrow$$Example: Roll control of rocket $$\mathcal{Q}$$ = aileron effectiveness, meaning, p.16-1 Time constant $$\tau$$

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R*3.14: Find h(x,y). (1)	P.17-1: If we select $$h(x,y) = k_2.$$ (constant), Then $$h_x = h_y = 0$$, and (1) is satisfied. Without assuming a priority that h = constant, discuss the search for the solution of (1). =Application: Generate Exact N2-ODEs:  Method: (reverse engineering)= 1.	Select $$\phi(x,y,p)$$ and set: $$\phi(x,y,p) = k$$	[[media:Pea1.f11.mtg16.djvu|Eqn(3) p.16-7]] Differentiate (3) p.16-7 to obtain the corresponding exact N2-ODE: 2.	$$G(x,y,y\prime,y\prime\prime) = \frac{d}{dx} \phi(x,y,y\prime) = 0$$	[[media:Pea1.f11.mtg16.djvu|Eqn(2) p.16-3]]

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Examples:

Exact N2-ODE: (2) p.16-3 -> [[media:Pea1.f11.mtg16.djvu|Eqns(2)-(4) p.16-4]]

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(3)-(8) p.18-2:

Pretend that $$\phi$$ is not known, show that the N2-ODE (1) is exact. 1st exactness condition for N2-ODEs(2) p.16-4: (1)	Clearly satisfies (2) p.16-4 with:

(1)	And (3) p.16-4:

2nd exactness condition for N2-ODEs(1)-(2) p.16-5: R*3.15: Verify whether (1) satisfies the 2nd exactness condition [[media:Pea1.f11.mtg16.djvu|Eqns(1)-(2) p.16-5]].

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Find the first integral $$\phi$$
 * 1) 4 p.16-4:

h(x,y) can be found using the definition of g(x,y,p). For (2) and (3) p.18-3:

The problem of finding h(x,y) in (3) is even more general than that in (1) p.18-1. Note: (3) is a PDE for h with (x,y) treated as independent variables. Similar to the two cases to solve (4) p.11-1 for the integrating factor for N1-ODEs, there are two possible choices (assumptions) that can be made.

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Note: Unlike the integrating factor h(x,y) in (4) p.11-1 for N1-ODEs, the function h(x,y) in (3) p.18-4 is not an integrating factor, but a function of the integration in (1) p.18-4. Yet, the solution procedure in both cases is similar. Remember Montesquieu p.3-2.