User:Egm6322.s12.team2.steele.m2/Mtg10

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

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Note: Smooth functions are functions that are infinitely differentiable, e.g., cos x, ex, log x, x, x2, etc. Examples of non-smooth functions: 1.	Heaviside (step) function The derivative is a generalized function: (1)

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2.	Hat (or roof) function (application in FEM): The derivative is a generalized function: The Hat function is not differentiable in the classical sense at $$(x_1, x_2, x_3)$$ (the “kinks”), i.e.,

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Limit on left of :  	$$\lim_{x \to x_i^-} \frac{df(x)}{dx}$$ Think: $$x_i^- = x_i - \epsilon$$, where $$\epsilon$$ is very small. Limit on right of $$x_i$$: $$\lim_{x \to x_i^+} \frac{df(x)}{dx}$$ Think: $$x_i^+ = x_i + \epsilon$$, where $$\epsilon$$ is very small. So, smooth -> no kinks (in any derivative order) (end NOTE) Summary: Two exactness conditions for N1-ODEs 1.	1st condition: Particular form [[media:Pea1.f11.mtg7.djvu|Eqn(2) p.7-6]] (see also p.8-6) 2.	2nd condition: [[media:Pea1.f11.mtg9.djvu|Eqn(3) p.9-3]]

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Example: See Example on p.8-6 Verify 2 exactness conditions for [[media:Pea1.f11.mtg8.djvu|Eqn(2) p.8-6]]. 1.	1st exactness condition (2) p.7-6: (2) p.8-6 satisfies $$M(x,y) + N(x,y) y\prime = 0$$		(2) p.7-6 2.	2nd exactness condition [[media:Pea1.f11.mtg8.djvu|Eqn(3) p.8-3]]: $$M_y(x,y) = 0$$		Thus (2) p.8-6 is exact $$N_x(x,y) = 0$$ Pretend that $$\phi(x,y)$$ was not known, and could be found from the above data (by some method) to be as given in (2) p.8-6. Then the solution y(x) is obtained by solving (2) p.7-6 for y in terms of x

Integration constant

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R*2.10: Verify that (1) p.10-4 is indeed the solution for the N1-ODE (2) p.8-6. Q: Suppose that an N1-ODE satisfies the 1st exactness condition (2) p.7-6, but not the 2nd exactness condition [[media:Pea1.f11.mtg9.djvu|Eqn(3) p.9-3]]. Is it possible to transform the original N1-ODE into an exact N1-ODE? A: yes, use the following … Euler Integrating factor method (IFM) Consider an N1-ODE with the particular form $$M(x,y) + N(x,y) y\prime = 0$$			[[media:Pea1.f11.mtg6.djvu|Eqn(2) p.6-6]]

i.e., satisfying the 1st exactness condition, but such that i.e., not satisfying the 2nd exactness condition (3) p. 8-3.

=Integrating Factor=

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Euler’s idea: Find an integrating factor h(x,y) such that the following N1-ODE is exact: