User:Egm6322.s12.team2.steele.m2/Mtg17

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

Page 17-0
Plan: Application, exact N2-ODE (p.16-7 cont’d) Note: Comparing exactness conditions for N1-ODE vs those of N2-ODEs 2nd exactness condition of N2-ODEs: Derivation

[[media:Pea1.f11.mtg17.djvu|Page 17-1]]
=Application (p.16-7 cont’d): = [[media:Pea1.f11.mtg16.djvu|Eqns(4)-(5) p.16-7]]:

The first integral in [[media:Pea1.f11.mtg16.djvu|Eqn(3) p.16-7]] becomes

Step 2: Solve the reduced-order N1-ODE (4) (by separation of variables):

[[media:Pea1.f11.mtg17.djvu|Page 17-2]]
2 integration constants for the N2-ODE [[media:Pea1.f11.mtg16.djvu|Eqn(1) p.16-6.]] Note: Recall N1-ODE; 1st exactness condition [[media:Pea1.f11.mtg7.djvu|Eqn(2) p.7-6:]] $$G(x,y,y\prime) = 0$$ is exact if $$\exists \phi(x,y)$$ such that

=2nd Exactness Condition [[media:Pea1.f11.mtg9.djvu|Eqn(3) p.9-3]]: =

Cf. [[media:Pea1.f11.mtg16.djvu|Eqns(1)-(2) p.16-5]]for N2-ODE Clairaut 1739-40

[[media:Pea1.f11.mtg17.djvu|Page 17-3]]
Then $$\phi(x,y) = k$$ is the 1st integral. 2nd exactness condition of N2-ODEs: Derivation 2nd relation (2) p.16-5: Easy, see R*3.13, Q1, p.16-6 1st relation (1) p.16-5: $$\phi(x,y,p)$$ has 3 arguments; hence 3 relations for the symmetry of mixed 2nd partial derivatives:

Circular permutation between x, y, and p Number [[media:Pea1.f11.mtg16.djvu|Eqn(3) p.16-4:]] $$g = \phi_x + \phi_y \, y\prime$$ Number [[media:Pea1.f11.mtg16.djvu|Eqn(4) p.16-4:]] $$f = \phi_p$$

[[media:Pea1.f11.mtg17.djvu|Page 17-4]]
Step 1: [[media:Pea1.f11.mtg17.djvu|Eqn(1c) p.17-3:]]

(1b) p.17-3 and (4) p.16-4:

Express $$\phi_y$$ in terms of (f,g,p):

[[media:Pea1.f11.mtg17.djvu|Page 17-5]]
Step 2: (1b) p.17-3 and (3)-(4) p.16-4:

With $$\phi_{xp} = \phi_{px} = f_x \, \phi_x$$ can be expressed in terms of $$(f,g,p)$$:

Step 3: Use (1a) p.17-3, i.e., $$\phi_{xy} = \phi_{yx}$$, and (5) p.17-4 and (3) to derive the 1st relation [[media:Pea1.f11.mtg16.djvu|Eqn(1) p.16-5]] of the 2nd exactness condition. (R*3.13, Question 2, p.16-6)