University of Florida/Egm6321/F10.TEAM1.WILKS/Mtg13

EGM6321 - Principles of Engineering Analysis 1, Fall 2009

Mtg 13: Tues, 22Sept09

Page 13-1

Euler integrating factor method Eq.(1) P.12-3

$\begin{array}{c}(x^m{y}^n)[\sqrt{x}{y}^{″}+2x{y}^{\prime }+3y]=0\end{array}$

Where: $(x^w{y}^n)=h(x,y)$

HW: Find (m,n) such that Eq(1) is exact.

Result: A first integration is:

$\begin{array}{c}\varphi (x,y,p)=xp+(2x^{\frac{3}{2}}-1)y+k_1=k_2\end{array}$

Where $p=y^{\prime }$ and $k_1$ and $k_2$ are constants

HW: Solve Eq(2) for $y(x)$ HINT: L1_ODE_VC (integrating factor)

A class of exact L2_ODE_VC (how to invent more exact L2_ODE_VC)

HW: Find mathematical structure of $\varphi$ that yields the above class

Page 13-2

$F=\frac{d\varphi }{dx}=\varphi {}_x(x,y,p)+\varphi {}_{y}p+\varphi {}_p{p}^{\prime }=P(x)y^{″}+Q(x)y^{\prime }+R(x)y$

Where $p=y^{\prime }$ and $p^{\prime }=y^{″}$

and $P(x)=\varphi {}_p$ and $Q(x)=\varphi {}_y$ and $R(x)y=\varphi {}_x$

Answer:

$\begin{array}{c}\varphi (x,y,p)=P(x)p+T(x)y+k\end{array}$

Eq.(2) P.13-1  : $P(x)=x$ and $T(x)=2x^{\frac{3}{2}}-1$ and $k=k_1$

Exact Nn_ODE's, where N means nonlinear and n means nth order

$\begin{array}{c}F(x,y^{(0)}...y^{(n)}=0\end{array}$

Where $y^{(0)}=y$ and $y^{(1)}=y^{\prime }$ and $y^{(n)}=\frac{d^{n}y}{d{x}^n}$

Page 13-3

Condition 1 for exactness:

$\begin{array}{c}F=\frac{d\varphi }{dx}(x,y^{(0)},...,y^{n-1})=\varphi {}_x+\varphi {}_{y(0)}{y}^{(1)}+...+\varphi {}_{y(n-1)}{y}^{(n)}\end{array}$

Where $\varphi {}_x+\varphi {}_{y(0)}{y}^{(1)}+...+\varphi {}_{y(n-1)}{y}^{(n)}$ is related term by term to $x,y^{(0)},...,y^{n-1}$

Condition 2 of exactness: $f_i:=\frac{\partial F}{\partial y^{(i)}}$, where $i=1,...,n$

$\begin{array}{c}{f}_0-\frac{d{f}_1}{dx}+\frac{d^2{f}_2}{d{x}^2}-...+(-1{)}^n\frac{d^n{f}_n}{d{x}^n}=0\end{array}$

HW: Case $n=1$ (N1_ODE)

$F(x,y,y^{\prime })=0=\frac{d}{dx}\varphi (x,y)$

$f_0-\frac{d{f}_1}{dx}=0\iff \varphi {}_{xy}=\varphi {}_{yx}$

HINT: $f_1=\varphi {}_y$

Page 13-4

Case $n=2$ (N2_ODE)

$f(x,y,y^{\prime },y^{″})=0=\frac{d\varphi }{dx}(x,y,y^{\prime })$

$f_0-\frac{d{f}_1}{dx}+\frac{d^2{f}_2}{d{x}^2}=0$

References

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