User:Egm6322.s12.team2.steele.m2/Mtg8

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

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Euler integrating factor method (cont’d p.7-6) Example: for [[media:Pea1.f11.mtg7.djvu|Eqn(2) p.7-6,]] a particular case of N1-ODE R*2.4: Verify that (3) is a N1-ODE.

Note: While (2) p.7-6 is in general an N1-ODE, it is still a particular N1-ODE, not the most general N1-ODE, which has the form [[media:Pea1.f11.mtg7.djvu|Eqn(1) p.7-6.]] Satisfied (1) p.7-6 but can be converted to (2) p.7-6 with the definition:

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Satisifies (1) p.7-6 but can be converted to (2) p.7-6 with Condition for (4) to hold If (5) is satisfied, then (4), which satisfies (1) p.7-6, can be converted to (2) p.7-6 with

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Note: Functions like M(x,y) and N(x,y) include constants (think of $$x^0 = y^0 = 1$$ ). Power zero Q: So when do we have a general N1-ODE of the form (1) p.7-6 that cannot be converted into the particular form (2) p.7-6? A.	Eqs. (3) – (4) p.8-1, and (2), (4) p.8-2 can all be expressed more generally as Where F(y’) is any function of y’ (nonlinear in general). If the function $$F(\cdot)$$ has no explicit inverse $$F^{-1}(\cdot)$$, then (2) cannot be put in the particular form (2) p.7-6.

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Definition: Exact N1-ODE A general N1-ODE of the form (1) p.7-6 is exact if there exists a function $$\phi(x,y)$$ such that

For any constant k, the level set

Leads to the N1-ODE (1) p.7-6: i.e. (4) is a first integral of (5).

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Note: Given a general N1-ODE (1) p.7-6, if a function  can be found such that (3)-(4) p.8-4 are satisfied, then we have effectively reduced the order of the N1-ODE (1) p.7-6 from 1 to 0, as represented by (4) p.8-4. The solution y(x) is then obtained if we can solve (4) p.8-4 for y in terms of x, with k being the integration constant.

Let’s carry out the total derivative in (4) p. 8-4 (remember the first total time derivative [[media:Pea1.f11.mtg2.djvu|Eqn(3) p.2-4,]] in high-speed train modeling and Montesquieu p.3-2) Which has the particular form (2) p.7-6 M(x,y) + N(x,y)y’ = 0 With

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=First Exactness Condition for N1-ODEs:= Thus the necessary condition for an N1-ODE to be exact is that it has the particular form (2) p.7-6. In other words, if an N1-ODE cannot be put in the particular form (2) p.7-6, then it cannot be exact. Example: Reverse engineering:  Given $$\phi (x,y)$$, find the N1-ODE with particular form (2) p.7-6.

M(x,y)		N(x,y)