User:Egm6322.s12.team2.steele.m2/Mtg21

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

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Plan: 2nd version of RTT: p.19-12 cont’d, R4.1 Non-exact L2-ODE-VC Example p.21-1 ODEs in power form: N1-ODE, L2-ODE-VC Integrating factor Non-exact L2-ODE-VC, integrating factor Example p.21-1 cont’d First integral for a class of exact L2-ODE-VC Example Application Exact Nn-ODE’s (Non-linear nth order) Case n=1: N1-ODE

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=Non-Exact L2-ODE-VC= Example: Consider the following ODE R*4.2: Verify the exactness of (1). Method (recipe) ODEs in special power form: $$a, b, c, d \in \mathbb R$$ $$p, q, r, s \in \mathbb R$$ $$r, s, t \in \mathbb R$$ $$\alpha, \beta, \gamma \in \mathbb R$$

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For ODEs of the form (2)-(3) p.21-1, use the integrating factor of the form

With $$m, n \in \mathbb R$$ to be determined.

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Non-exact L2-ODE-VC, integrating factor Example P.21-1 cont’d L2-ODE-VC (1) p.21-1 and integrating factor (1) p.21-2. R*4.3: Find $$m, n \in \mathbb R$$ such that the following N2-ODE is exact: Show that the first integral is a L1-ODE-VC

With $$p(x) := y\prime(x)$$	[[media:Pea1.f11.mtg7.djvu|Eqn(2) p.7-3]] Solve (2) for y(x). (end R*4.3) /// (end Example) ///

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First integral for a class of exact L2-ODE-VC $$\phi(x,y,p)$$	First integral Recall [[media:Pea1.f11.mtg16.djvu|Eqns(2)-(3) p.16-3:]] Consider the L2-ODE-VC:

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The following function generates a class of exact L2-ODE-VC: See F09 p.13-1 for the problem statement on the derivation of (1). R*4.4: Show that (2)-(3) p.21-4 leads to (1). Example p.21-3

Application Select $$\phi$$ satisfying (1):

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R*4.5: L2-ODE-VC

1.	Show (5) is exact. 2.	Find $$\phi$$ 3.	Solve for y(x) 1.	(end R*4.5)		/// 2.	(end Application)	/// 3.

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=Exact Nn-ODE’s (Non-linear nth order)= $$G(x,y^{(0)}, y^{(1)}, y^{(2)}, \cdots, y^{(n)}) = 0$$ [[media:Pea1.f11.mtg7.djvu|Eqn(1) p.7-2]] $$\displaystyle y^{(k)} (x) := \frac{d^ky(x)}{dx^k}$$	. [[media:Pea1.f11.mtg4.djvu|Eqn(2) p.4-6]] Exactness Condition 1:

Functions of $$(x, y^{(0)}, y^{(1)}, \cdots, y^{(n-1)})$$ Exactness Condition 2: Partial derivative

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Total derivatives

Equivalently in more compact form: Case n=1: N1-ODE 2nd exactness condition (1)-(2): Which is equivalent to the 2nd exactness condition for N1-ODE in [[media:Pea1.f11.mtg9.djvu|Eqns(2)-(3) p.9-3:]] $$\phi_{xy} = \phi_{yx} \Leftrightarrow M_y = N_x$$		(2)-(3) p.93