University of Florida/Egm6341/s11.TEAM1.WILKS/Mtg11

EGM6321 - Principles of Engineering Analysis 1, Fall 2010

Mtg 1: Thur,24Aug10

Page 11-1

HW P.10-4 (continued)
2) Assume $a_{1}(x)\neq 0\forall \ x\$ , Eq(8) P.10-3 becomes $y'+{\frac {a_{0}(x)}{a_{1}(x)}}y={\frac {b(x)}{a_{1}(x)}}\$

Where ${\frac {a_{0}(x)}{a_{1}(x)}}=P(x)\$ and ${\frac {b(x)}{a_{1}(x)}}=Q(x)\$ from K.p.512

Find expression for $y(x)\$ in terms of $a_{0},a_{1},b\$.

3) $a_{1}(x)=x^{2}+1\$
$b(x)=2x\$
$a_{0}(x)=x\$

NOTE: cf. to K.p.512

1) K. etal. did not derive expression Eq.(1)p.10-3 $h(x)=e^{\left[\int _{}^{x}a_{0}(s)\,ds\right]}\$

"pulling rabbit out of hat"

2) $\oint _{}^{x}f(s)\,ds:=\int _{}^{x}f(s)\,ds\ \equiv \int f(x)\,dx$ without constant in K.2003

Page 11-2

Lecture: $\int _{}^{x}f(s)\,ds=\int f(x)\,dx+k=\oint _{}^{x}f(s)\,ds+k\$

Eq.(6)p.10-3 :2 constants $k_{1}\$ and $k_{2}\$

Eq.(1)p.10-3 : $h(x)\rightarrow \ k_{1}\$

Eq.(6)p.10-3 : $\int _{}^{x}h(s)b(s)\,ds\rightarrow \ k_{2}\$

But Eq.(5)p.10-2 is L1_ODE_VC

HW: $\alpha \ \$ Show that $k_{1}\$ is not necessary.

HW: $\beta \ \$ Show Eq.(6)p.10-3 agrees with K.p.512, i.e. $y(x)=Ay_{H}(x)+y_{P}(x)\$

HW: $\gamma \ \$ Find $y_{H}(x)\$ independant, i.e. solve $y'+a_{0}y=0\$

$\delta \ \$ How about $y_{P}(x)\$ ? $\Rightarrow \ \$ Variation fo parameters (later)

Page 11-3

A class of exact N1_ODE:

Recall Eq.(7)p.10-1 (Case 1)
One possibility to satisfy this condition: Consider:

$\displaystyle {\begin{aligned}N(x,y)=N(x)\end{aligned}}$

(1)

$\displaystyle {\begin{aligned}N_{x}(x,y)=b(x)\end{aligned}}$

(2)

$\displaystyle {\begin{aligned}M_{y}(x,y)=a(x)\end{aligned}}$

(3)

$\displaystyle {\begin{aligned}\Rightarrow \ M(x,y)=a(x)y+k(x)\end{aligned}}$

(4)

$\displaystyle {\begin{aligned}N(x,y)=\int _{}^{x}b(s)\,ds=:{\bar {b}}\ (x)\end{aligned}}$

(5)

$\displaystyle {\begin{aligned}M+Ny'=\left[a(x)y+k(x)+{\bar {b}}\ (x)y'=0\right]\end{aligned}}$

(6)

Where Eq(6) is a L1_ODE_VC (not necessarily exact, but can be made exact: integrating factor method)

Application: Consider $a(x)=x^{4}\neq \ b(x)=x\Rightarrow \ {\bar {b}}\ (x)={\frac {1}{2}}x^{2}\$
$k=10\$

$\displaystyle {\begin{aligned}\left[x^{4}y+10\right]+\left({\frac {1}{2}}x^{2}\right)y'=0\end{aligned}}$

(7)

F09: Find $h(x)\$ such that Eq.(7) is exact

Page 11-4

Question: But Eq.(6)p.11-3 is linear!
Find N1_ODEs that are exact or can be made exact by integrating factor method.