# 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 \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_0y=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{align} N(x,y)=N(x) \end{align} (1)
 \displaystyle \begin{align} N_x(x,y)=b(x) \end{align} (2)
 \displaystyle \begin{align} M_y(x,y)=a(x) \end{align} (3)
 \displaystyle \begin{align} \Rightarrow \ M(x,y)=a(x)y+k(x) \end{align} (4)
 \displaystyle \begin{align} N(x,y)=\int_{}^{x} b(s)\, ds =: \bar b \ (x) \end{align} (5)
 \displaystyle \begin{align} M+Ny'= \left [ a(x)y+k(x)+ \bar b \ (x)y'=0 \right ] \end{align} (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 \ne \ b(x)= x \Rightarrow \ \bar b \ (x) = \frac{1}{2}x^2 \$
$k=10 \$

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