University of Florida/Egm6321/F10.TEAM1.WILKS/Mtg9

EGM6321 - Principles of Engineering Analysis 1, Fall 2009

Mtg 9: Thur, 10Sept09

Page 9-1

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

Note: Symbol notations:

$\mathrm{\Delta }$ defined as equal by definition

Non-symmetric notation $:=$ means equal by definition as well, better notation than $\mathrm{\Delta }$

Goal: Derive mathematical structure of a class of exact N1_ODEs

Exact L1_ODE_VC:

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

Application: Just invent any $a(x),b(x)$
Let $a(x)=x^4$

$b(x)=x\Rightarrow \overline{b}(x)=\frac{1}{2}{x}^2$

$k=10$

Page 9-2

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

HW: Show that Eq(3) L1_ODE_VC is "exact". See Note P.10-1

Question: How about N1_ODEs? Eq.(2)P.6-4

N1 means Nonlinear, 1st Order

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

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

Where $\underset{}{\overset{x}{\int }}b(s)=\overline{b}(x)$

Eq.(2) P.6-4:

$\frac{1}{N}(N_x-M_y)=\frac{1}{\overline{b}(x)c(y)}[b(x)c(y)-a(x)c(y)]=-f(x)$

Page 9-3

Eq.(4) P.9-2 $M(x,y)=a(x)\underset{}{\overset{y}{\int }}c(s)$

Where $\underset{}{\overset{y}{\int }}c(s)=\overline{c}(y)$

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

From Eq.(6) P.9-3 , Eq.(4) P.9-2 and Eq.(3) P.4-2 obtain:

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

Application: Consider the following

$a(x)=5x^3+2$

$b(x)=x^2\Rightarrow \overline{b}(x)=\frac{1}{3}{x}^3$

$c(y)=y^4\Rightarrow \overline{c}(y)=\frac{1}{5}{y}^5$

$\begin{array}{c}\left(\frac{1}{3}{s}^3\right)\left(y^4\right)y^{\prime }+(5x^3+2)\left(\frac{1}{5}{y}^5\right)=0\end{array}$

HW: Show Eq(8) is "exact" N1_ODE. Note P.10-1

Page 9-4

L2_ODE_VC with missing dependant variable

$P(x)y^{″}+Q(x)y^{\prime }+R(x)y=S(x)$

Where $R(x)\to 0$ due to missing dependant variable y.

Eq.(1)P.2-3 $p:=y^{\prime }\to P(x)p^{\prime }+Q(x)p=S(x)$ is a L1_ODE_VC

Solution: Eq.(4)P.8-2

Exact N2_ODEs:

General N2_ODEs: $F(x,y,y^{\prime },y^{″})=0$

Application: $(x^3+2x^5{y}^2)y^{″}+(x^{\frac{3}{2}}+10)\sqrt{y}{y}^{\prime }+y^{100}=0$

$F=0$ is exact means $\mathrm{\exists }\varphi (x,y,y^{\prime })$ such that $F(x,y,y^{\prime },y^{″})=\frac{d}{dx}\varphi (x,y,y^{\prime })$

References

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