# University of Florida/Egm6321/F10.TEAM1.WILKS/Mtg21

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

Mtg 21: Thurs, 8Oct09

### Page 21-1

P.20-4 (continued)

$\Rightarrow ax^{-1}\$  is a homgenous solution $\forall \!\,a\$

$u_{1}(x)=c_{1}x^{-1}\$

2) $b=2\Rightarrow ax^{2}\$  is another homogeneous solution since $b^{2}-b-2=0\$

$u_{2}(x)=c_{2}x^{2}\$

(Verify $u_{1}\$  and $u_{2}\$  are linearly independant components of $W\$

3) $b=6,a={\frac {1}{4}}\Rightarrow \$  left hand side of Eq(1) p20-4 $=7x^{4}\$  , where $=7x^{4}\$  is the 1st term on the right hand side

for $b=6,a={\frac {1}{4}}\Rightarrow {\frac {1}{4}}x^{6}\$

4) $b=5,a={\frac {1}{6}}\Rightarrow \$  left hand side of Eq(1) p20-4 $=3x^{3}\$  , where $=3x^{3}\$  is the 2nd term on the right hand side

for $b=5,a={\frac {1}{6}}\Rightarrow {\frac {1}{6}}x^{5}\$

### Page 21-2

Llinearity of ordinary differential equation $\Rightarrow \$  superposition

$y_{P}(x)={\frac {1}{4}}x^{6}+{\frac {1}{6}}x^{5}\$

$y(x)=c_{1}x^{-1}+c_{2}x^{2}+y_{P}(x)\$  , where $c_{1}x^{-1}+c_{2}x^{2}=y_{H}(x)\$

Alternative method to obtain full solution for non-homogeneous L2_ODE_VC knowing only one homogeneous solution (e.g. obtained by trial solution) (bypassing reduction of order method2-undertermined factor for $u_{2}\$  and variation of parameter method)

Eq.(1) P.3-1 = $y''+a_{1}(x)y'+a_{0}(x)y=f(x)\$

Assume having found $u_{1}(x)\$ , a homogeneous solution: $u_{1}''+a_{1}(x)u_{1}'+a_{0}(x)u_{1}=0\$

Consider: $y(x)=U(x)u_{1}(x)\$  , where $U(x)\$  is an undetermined factor

### Page 21-3

Follow the same argument as on P.17-2 to obtain:

 \displaystyle {\begin{aligned}f(x)=U'(a_{1}u_{1}+2u_{1}')+U''u_{1}\end{aligned}} (1)

NOTE: this equation is missing the dependant variable $U\$  in front of $U'\$  term due to reduction of order method $\phi \ \$

 \displaystyle {\begin{aligned}Z(x):=U'(x)\end{aligned}} (2)

 \displaystyle {\begin{aligned}\Rightarrow u_{1}(x)Z'+\left[a_{1}(x)u_{1}(x)+2u_{1}'(x)\right]Z=f(x)\end{aligned}} (3)

where $u_{1}(x)\$  and $\left[a_{1}(x)u_{1}(x)+2u_{1}'(x)\right]\$  are known

Non-homogeneous L1_ODE_VC solution for $Z(x)\$  : Eq.(4) P.8-2

 \displaystyle {\begin{aligned}U(x)=\int _{}^{x}Z(s)\,ds\end{aligned}} (4)

 \displaystyle {\begin{aligned}y(x)=U(x)u_{1}(x)\end{aligned}} (5)

ref: K p.28, problem 1.1ab

a) $u_{1}(x)=e^{x}\$  , $(x-1)y''-xy'+y=0\$

Trial solution $y(x)=e^{rx}\$  , where $r=\$  constant

### Page 21-4

Find $r_{1},r_{2}\$

How many valid homogeneous solutions to $u_{1}=e^{r_{1}x}\$ , find $u_{2}\$  using undetermined factor method