User:Egm6321.f09.mtg3

=EGM6321 - Principles of Engineering Analysis 1, Fall 2009= MEETING 3 - Thursday, 27Aug09

3-1
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 p2.3, Eq(1): If $$ P(x)\ne \ 0 \ \forall x$$, divide throughout by $$ P(x) \ $$ to get: EQ1) $$ 1y''+\frac{Q}{P}y'+\frac{R}{P}y=\frac{F}{P} $$ 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

3-2
2nd order--> need 2 conditions to solve for 2 constraints Boundary Value Problem (BVP) Prescribe: (1) $$ y(a)=\alpha\ \ $$ (1) $$ y(b)=\beta\ \ $$ where $$ \alpha\ \ $$ and $$ \beta\ \ $$ are known values Initial Value Problem (IVP) Prescribe: (2) $$ y(a)=\alpha\ \ $$ (2) $$ y'(a)=\beta\ \ $$ 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

3-3
2) Superposition based on linearity of differential operation L(.) (1) $$ L_2(.) = \frac{d^2(.)}{dx^2} + a_1\frac{d(.)}{dx}+a_0(.) $$ Where the 2 in $$ L_2(y) \ $$ is defined as 2nd order (2) $$ L_2(y)  = y'' + a_1y'+a_0y \ $$ (3) Linearity of  $$ L(.) \ $$: $$ \forall u,v \ $$ 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 $$ \mathbf{A} \epsilon\ \mathbb R \ ^{nxm} $$ matrix with n rows and m columns of real numbers $$\forall \mathbf{u}, \mathbf{v} \epsilon\ \mathbb R \ ^{mx1} \ $$ is a column matrix $$ \forall \alpha\, \beta\, \epsilon\ \mathbb R \ \ $$

3-4
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.