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

\begin{matrix} c_{1} ({u_{1}}^{″} + a_{1} {u_{1}}^{'} + a_{0} u_{1}) + c_{2} ({u_{2}}^{″} + a_{1} {u_{2}}^{'} + a_{0} u_{2}) + {c_{1}}^{'} {u_{1}}^{'} + {c_{2}}^{'} {u_{2}}^{'} = f \end{matrix}

Template:Font

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 ${\begin{matrix} {c_{1}}^{'} \\ {c_{2}}^{'} \end{matrix}}$

In matrix form: $[\begin{matrix} u_{1} & u_{2} \\ {u_{1}}^{'} & {u_{2}}^{'} \end{matrix}] {\begin{matrix} {c_{1}}^{'} \\ {c_{2}}^{'} \end{matrix}} = {\begin{matrix} 0 \\ f \end{matrix}}$

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

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

$W = d e t \underline{W}$

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

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