User:EGM6321.F10.TEAM1.WILKS/Mtg3

=EGM6321 - Principles of Engineering Analysis 1, Fall 2009=

Mtg 3: Thur, 27 Aug 09

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To find class homepage, go to Wikiversity: Main Page (http://en.wikiversity.org/wiki/Wikiversity:Main_Page) and search for--> user:egm6321.f09

My Wiki address is: http://clesm.mae.ufl.edu/wiki.vq/index.php/Main_Page

From [[media: Egm6321.f09.mtg2.djvu | Eq.(1)p.2-3]]: If $$ P(x)\ne \ 0 \ \forall x$$, divide throughout by $$ P(x) \ $$ to get:

Where:

$$\forall \ $$ is defined as "for all"

$$ a_2(x)=1 \ $$,

$$ a_1(x)=\frac{Q}{P} $$,

$$ a_0(x)=\frac{R}{P} $$, and

$$ f(x)=\frac{F}{P} $$

$$ \forall x_0 \ $$ such that $$ P(x_0) \ne\ 0 \ $$ then $$ x_0 \ $$ is a regular point

Any $$ x_0 \ $$ such that $$ P(x_0)=0 \ $$ is a regular point



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2nd order--> need 2 conditions to solve for 2 constraints

Boundary Value Problem (BVP)

Prescribe:

where $$ \alpha\ \ $$ and $$ \beta\ \ $$ are known values

Initial Value Problem (IVP)

Prescribe:

where $$ \alpha\ \ $$ and $$ \beta\ \ $$ are known values

Solve IVP by ODE from p3-1 Eq(1) or initial condition p3-2 Eq(2)

Two points:

1) Existence and uniqueness of solution



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2) Superposition based on linearity of differential operation L(.)

Where the 2 in $$ L_2(y) \ $$ is defined as 2nd order

$$ \forall u,v \ $$ in a function of x

and $$ \forall \alpha\, \beta\ \ $$ belonging to $$ \mathbb R \ $$ (scalars, real numbers);

$$ L( \alpha\ u+ \beta\ v)= \alpha\ L(u)+ \beta\ L(v) \ $$

Where $$ \mathbb R \ $$ is defined as a set of real numbers Example: Matrix Algebra

$$ \mathbf{A} \in \mathbb R \ ^{nxm} $$ matrix with n rows and m columns of real numbers

$$\forall \mathbf{u}, \mathbf{v} \in \mathbb R \ ^{mx1} \ $$ is a column matrix

$$ \forall \alpha\, \beta\, \in \mathbb R \ \ $$

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Clearly: $$ \mathbf{A}(\alpha\ \mathbf{u} + \beta\ \mathbf{v}) = \alpha\ \mathbf{A}\mathbf{u} + \beta\ \mathbf{A} \mathbf{v} $$

Example:

$$ \frac{d}{dx}(.) \ $$ is a linear operation

$$ (\alpha\ u+ \beta\ v)'= \alpha\ u'+\beta\ v' \ $$

linearity allows the use of superposition

$$ y=y_H+y_P \ $$

$$ L(y) = L(y_H) + L(y_P) \ $$, where the subscripts H and P stand for homogeneous and particular in respective order.