User:Egm6322.s12.team2.steele.m2/Mtg9

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

[[media:Pea1.f11.mtg9.djvu|Page 9-1]]
Note: Linearly-independent functions, p.6-4 1)	Case of vectors (restricted to $$\mathbb R^2$$ for simplicity) Linearly dependent		Linearly independent 2)	Case of functions R*2.6: The homogeneous solutions $$y^1_H(x)$$ in [[media:Pea1.f11.mtg7.djvu|Eqn(3) p.7-1]] and $$y^2_H(x)$$ in [[media:Pea1.f11.mtg7.djvu|Eqn(4) p.7-1]]are linearly independent, i.e., show that i.e. for any given $$\alpha$$, show that

[[media:Pea1.f11.mtg9.djvu|Page 9-2]]
Plot $$y^1_H(x)$$ and $$y^2_H(x)$$ HW: Read King 2003, Appendix 5, ODEs (particular cases of lectures), p.511-516. For example, compare $$g(y) \frac{dy}{dt} = f(t)$$		King 2003 p.511(A5.1) Or translated into our notation as To the particular form [[media:Pea1.f11.mtg6.djvu|Eqn(2) p.6-6]] $$M(x,y) + N(x,y) y\prime = 0$$ R*2.7: Consider the following function

[[media:Pea1.f11.mtg9.djvu|Page 9-3]]
And show that (3) p.9-2 is an N1-ODE. R*2.8 Does the following N1-ODE satisfy the first exactness condition? Hint: See [[media:Pea1.f11.mtg8.djvu|Eqn(2) p.8-3]] and [[media:Pea1.f11.mtg8.djvu|Eqn(2) p.8-4]].

Q: If $$\phi (x,y)$$ exists, what is then the relationship between$$M(x,y) = \phi_x (x,y) \ and \ N(x,y) = \phi_x (x,y)$$ ? Can M(x,y) and N(x,y) be chosen arbitrarily? NO. Assuming that  is smooth, we have: [[media:Pea1.f11.mtg8.djvu|Eqn(2) p.8-5]] and [[media:Pea1.f11.mtg8.djvu|Eqn(1) p.8-6]] ->    $$M_y(x,y)$$  $$N_x(x,y)$$

=Second Exactness Condition:=

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R*2.9 (PEA2 S09) Review calculus, and find the minimum degree of differentiability of the function $$\phi (x,y)$$ such that (2) p.9-3 is satisfied. State the full theorem and provide a proof.