User:Egm6322.s12.team2.steele.m2/Mtg12

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

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R*2.14: p.11-5 Cont’d 2.	Find y(x) in terms of $$a_x, \ a_0(x), \ b(x).$$ 3.	3. $$a_1(x) = x^2 + 1$$ $$a_0(x) = x$$ $$b(x) = 2x$$ Note: 1.	Cf. King 2003, Appendix 5, p.512, where the particular L1-ODE-VC in [[media:Pea1.f11.mtg11.djvu|Eqn(3) p.11-3]] was considered: $$y\prime + P(x) y = Q(x)$$ “If we multiply through by$$\exp{\int^x P(t)dt}$$, where …” 2.	King 2003 did not derive the integrating factor [[media:Pea1.f11.mtg11.djvu|Eqn(3) p.11-4]]; it was like pulling a rabbit out of a hat.

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R*2.15: Since (2)-(3) p.11-3 is an L1-ODE-VC, there should be only one integration constant, not two. Show that the integration constant $$k_1$$ in (3) p.11-4 is not necessary, i.e., only $$k_2$$ in (1) p.11-5 is necessary. R*2.16: Show that the solution of (3) p.11-3 in [[media:Pea1.f11.mtg11.djvu|Eqn(1) p.11-5]] agrees with the result presented in King 2003 p.512, i.e., Use (3) p.11-4 and (1) p.11-5 to identify A, $$y_H(x)$$ and $$y_P(x)$$. Compare your results with those in King 2003 p.512. R*2.17: Instead of identifying $$y_H(x)$$ from (3) p.11-4 and (1) p.11-5, solve the homogeneous counterpart of (3) p.11-3, i.e.,

How about ? Variation of parameters (soon).

=A Class of Exact N1-ODEs=

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Recall the condition [[media:Pea1.f11.mtg11.djvu|Eqn(2) p.11-2]] for Case 1 $$(h_y(x,y) = 0)$$ ( To satisfy the above condition, consider (1): N(x,y) = N(x) $$\Rightarrow k_1(y) = k_1, \bar b(x,y) = \bar b(x)$$

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Which is a particular L1-ODE-VC (not necessarily exact, but can be made exact by using the IFM). Application: Consider

i.e., select $$k_1 = 0$$ Also select $$k_2(x) = 10$$, so (1) becomes Which is a L1-ODE-VC that is either exact, or can be made exact by using the IFM.

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R*2.18: If (4) p.12-4 is not exact, find the integrating factor h to make it exact. But (1) p.12-4 is a class of general L1-ODE-VC that are either exact, or can be made exact by the IFM. Q: Is it possible to exhibit a class of N1-ODEs that are either exact, or can be made exact by the IFM?