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 $a_1(x)\ne 0\mathrm{\forall }x$ , Eq(8) P.10-3 becomes $y^{\prime }+\frac{a_0(x)}{a_1(x)}y=\frac{b(x)}{a_1(x)}$

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 $h(x)=e^{[\underset{}{\overset{x}{\int }}{a}_0(s)ds]}$

"pulling rabbit out of hat"

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

Page 11-2

Lecture: $\underset{}{\overset{x}{\int }}f(s)ds=\int f(x)dx+k=\underset{}{\overset{x}{\oint }}f(s)ds+k$

Eq.(6)p.10-3 :2 constants $k_1$ and $k_2$

Eq.(1)p.10-3 : $h(x)\to k_1$

Eq.(6)p.10-3 : $\underset{}{\overset{x}{\int }}h(s)b(s)ds\to k_2$

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. $y(x)=A{y}_H(x)+y_P(x)$

HW: $\gamma$ Find $y_H(x)$ independant, i.e. solve $y^{\prime }+a_{0}y=0$

$\delta$ 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:

$\begin{array}{c}N(x,y)=N(x)\end{array}$

$\begin{array}{c}{N}_x(x,y)=b(x)\end{array}$

$\begin{array}{c}{M}_y(x,y)=a(x)\end{array}$

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

$\begin{array}{c}N(x,y)=\underset{}{\overset{x}{\int }}b(s)ds=:\overline{b}(x)\end{array}$

$\begin{array}{c}M+N{y}^{\prime }=[a(x)y+k(x)+\overline{b}(x)y^{\prime }=0]\end{array}$

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

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

$\begin{array}{c}[x^{4}y+10]+\left(\frac{1}{2}{x}^2\right)y^{\prime }=0\end{array}$

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.

References

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