# University of Florida/Egm6321/F10.TEAM1.WILKS/Mtg3

EGM6321 - Principles of Engineering Analysis 1, Fall 2009

Mtg 3: Thur, 27 Aug 09

### Page 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

From Eq.(1)p.2-3: If $P(x)\neq \ 0\ \forall x$ , divide throughout by $P(x)\$ to get:

 \displaystyle {\begin{aligned}1y''+{\frac {Q}{P}}y'+{\frac {R}{P}}y={\frac {F}{P}}\end{aligned}} (1)

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})\neq \ 0\$ then $x_{0}\$ is a regular point

Any $x_{0}\$ such that $P(x_{0})=0\$ is a regular point

### Page 3-2

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

Boundary Value Problem (BVP)

Prescribe:

 \displaystyle {\begin{aligned}y(a)=\alpha \ \end{aligned}} (1)

 \displaystyle {\begin{aligned}y(b)=\beta \ \end{aligned}} (1)

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

Initial Value Problem (IVP)

Prescribe:

 \displaystyle {\begin{aligned}y(a)=\alpha \ \end{aligned}} (2)

 \displaystyle {\begin{aligned}y'(a)=\beta \ \end{aligned}} (2)

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

### Page 3-3

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

 \displaystyle {\begin{aligned}L_{2}(.)={\frac {d^{2}(.)}{dx^{2}}}+a_{1}{\frac {d(.)}{dx}}+a_{0}(.)\end{aligned}} (1)

 \displaystyle {\begin{aligned}L_{2}(y)=y''+a_{1}y'+a_{0}y\end{aligned}} (2)

Where the 2 in $L_{2}(y)\$ is defined as 2nd order

 Linearity of \displaystyle {\begin{aligned}L(.)\end{aligned}} (3)
$\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} \ \$ ### Page 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.