# University of Florida/Egm6321/F10.TEAM1.WILKS/Mtg19

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

Mtg 19: Tues, 5Oct09

### Page 19-1

HW: Legendre differential Eq.(1) P.14-2 with ${\displaystyle n=0\ }$ , such that homogeneous solution ${\displaystyle u_{1}(x)=1\ }$ .

Use reduction of order method 2 (undetermined factor) to find ${\displaystyle u_{2}(x)\ }$ , second homgenous solution

HW: K. p28, pb. 1.1.b.

Variation of parameters (continued) P.18-4

Use expression for ${\displaystyle y'\ }$  Eq.(2) P.18-4 and ${\displaystyle y''\ }$  Eq.(3) P.18-4 in non-homogeneous L2_ODE_VC Eq.(1) P.3-1

 {\displaystyle \displaystyle {\begin{aligned}c_{1}{\cancel {(u_{1}''+a_{1}u_{1}'+a_{0}u_{1})}}+c_{2}{\cancel {(u_{2}''+a_{1}u_{2}'+a_{0}u_{2})}}+c_{1}'u_{1}'+c_{2}'u_{2}'=f\end{aligned}}} (1)

Where ${\displaystyle u_{1}''+a_{1}u_{1}'+a_{0}u_{1}\Rightarrow 0\ }$  , because ${\displaystyle u_{1}\ }$  is a homogeneous solution

Where ${\displaystyle u_{2}''+a_{1}u_{2}'+a_{0}u_{2}\Rightarrow 0\ }$  , because ${\displaystyle u_{2}\ }$  is a homogeneous solution

### Page 19-2

2 equations Eq.(1) P.18-4 and Eq.(1) P.19-1 for two unknowns ${\displaystyle {\begin{Bmatrix}c_{1}'\\c_{2}'\end{Bmatrix}}\ }$

In matrix form: ${\displaystyle {\begin{bmatrix}u_{1}&u_{2}\\u_{1}'&u_{2}'\end{bmatrix}}{\begin{Bmatrix}c_{1}'\\c_{2}'\end{Bmatrix}}={\begin{Bmatrix}0\\f\end{Bmatrix}}\ }$

Where ${\displaystyle {\begin{bmatrix}u_{1}&u_{2}\\u_{1}'&u_{2}'\end{bmatrix}}\ }$  is the Wronskian matrix designated as ${\displaystyle {\underline {W}}\ }$

The Wronskian, W, is the determinant of ${\displaystyle {\underline {W}}\ }$

${\displaystyle W=det{\underline {W}}\ }$

If ${\displaystyle W\neq \ 0\ }$ , then ${\displaystyle {\underline {W}}^{-1}\ }$  exists and ${\displaystyle {\begin{Bmatrix}c_{1}'\\c_{2}'\end{Bmatrix}}={\underline {W}}^{-1}{\begin{Bmatrix}0\\f\end{Bmatrix}}\ }$

Theorem: ${\displaystyle \forall \!\,}$  ${\displaystyle u_{1},u_{2}\ }$  (function of x) are linearly independant if ${\displaystyle {\underline {W}}\neq \ 0\ }$  , where ${\displaystyle 0=\ }$  zero function.

### Page 19-3

 {\displaystyle \displaystyle {\begin{aligned}{\underline {W}}^{-1}={\frac {1}{W}}{\begin{bmatrix}u_{2}'&-u_{2}\\-u_{1}'&u_{1}\end{bmatrix}}\end{aligned}}} (1)

 {\displaystyle \displaystyle {\begin{aligned}{\begin{Bmatrix}c_{1}'\\c_{2}'\end{Bmatrix}}={\underline {W}}^{-1}{\begin{Bmatrix}0\\f\end{Bmatrix}}={\frac {1}{W}}{\begin{Bmatrix}-u_{2}f\\u_{1}f\end{Bmatrix}}\end{aligned}}} (2)

Where ${\displaystyle {\begin{Bmatrix}-u_{2}f\\u_{1}f\end{Bmatrix}}\ }$  are known

 {\displaystyle \displaystyle {\begin{aligned}c_{1}(x)=\int _{}^{x}{\frac {u_{2}(s)f(s)}{W_{s}}}\,ds+A\end{aligned}}} (3)

Where ${\displaystyle \int _{}^{x}{\frac {u_{2}(s)f(s)}{W_{s}}}\,ds=d_{1}(x)\ }$

 {\displaystyle \displaystyle {\begin{aligned}c_{2}(x)=\int _{}^{x}{\frac {u_{1}(s)f(s)}{W_{s}}}\,ds+B\end{aligned}}} (4)

Where ${\displaystyle \int _{}^{x}{\frac {u_{1}(s)f(s)}{W_{s}}}\,ds=d_{2}(x)\ }$