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 $n=0\$, such that homogeneous solution $u_{1}(x)=1\$.

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

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

Variation of parameters (continued) P.18-4

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

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

(1)

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

Where $u_{2}''+a_{1}u_{2}'+a_{0}u_{2}\Rightarrow 0\$ , because $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 ${\begin{Bmatrix}c_{1}'\\c_{2}'\end{Bmatrix}}\$

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

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

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

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

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

Page 19-3

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

(1)

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

(2)

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

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

(3)

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

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

(4)

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