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

EGM6321 - Principles of Engineering Analysis 1, Fall 2011

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 Eqn(3) p.7-1 and $y^2_H(x)$ in Eqn(4) p.7-1are linearly independent, i.e., show that

$\forall \alpha \in \mathbb R , \ y^1_H(\cdot) \ne \alpha \, y^2_H(\cdot)$

$\displaystyle \color{red}(1)$

i.e. for any given $\alpha$, show that

$\exists \hat x \ {\rm such \ that \ } \ y^1_H(\hat x) \ne \alpha y^2_H(\hat x)$

$\displaystyle \color{red}(2)$

 

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

$M(x) + N(y) \, y\prime = 0$

$\displaystyle \color{red}(1)$

To the particular form Eqn(2) p.6-6
$M(x,y) + N(x,y) y\prime = 0$
R*2.7: Consider the following function

$\phi (x,y) = x^2y^{\frac{3}{2}} + log(x^3y^2) = k$

$\displaystyle \color{red}(2)$

Find $G(y\prime,y,x) = \frac{d}{dx} \phi(x,y) = 0$

$\displaystyle \color{red}(3)$

 

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 Eqn(2) p.8-3 and Eqn(2) p.8-4.

$M(x,y) \cos y\prime + N(x,y) \log y\prime = 0$

$\displaystyle \color{red}(1)$

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:

$\phi_{xy}(x,y) = \phi_{yx}(x,y)$ i.e., $\frac{\partial^2 \phi (x,y)}{\partial x \partial y} = \frac{\partial^2 \phi (x,y)}{\partial y \partial x}$

$\displaystyle \color{red}(2)$

Eqn(2) p.8-5 and Eqn(1) p.8-6 -> $M_y(x,y)$ $N_x(x,y)$

Second Exactness Condition:

$M_y(x,y) = N_x(x,y)$ i.e., $\frac{\partial M(x,y)}{\partial y} = \frac{\partial N(x,y)}{\partial x}$

$\displaystyle \color{red}(3)$

 

Page 9-4

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.