EGM6321 - Principles of Engineering Analysis 1, Fall 2009
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Mtg 5: Thur, 03 Sept 09
Note: Eq.(2)p.4-2 and Eq.(4)p.4-2
Integrate from a to x:
, where
,
another way:
Where
and
But
|
(1)
|
Eq.(3)p.4-2 : Why this form of nonlinear 1st order ODE?
Most general form:
|
(2)
|
Application:
|
(3)
|
Where
is defined as
HW: Show that
in Eq(3) is a nonlinear 1st order ODE.
Hint: Define the differential operator
associated with Eq(3).
Form for exact nonlinear 1st order ODE:
is exact if
a function
such that
|
(1)
|
where:
is defined as "there exists"
Multiply Eq1) thru by
to get:
Eq.(3)p.4-2 :
NOTE: If
does not have the form:
|
(2)
|
Then
cannot be exact.
Application: Eq.(3)p.5-2 is not exact because of the nonlinear term
Exactness test (continued)
p5-3 Eq(1):
Since
and ![{\displaystyle \phi \ _{xy}={\frac {\partial ^{2}\phi \ }{\partial x\partial y}}\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/09524808dd26b51484dfe55aa46f3404e8ca9bfd)
and ![{\displaystyle \phi \ _{yx}={\frac {\partial ^{2}\phi \ }{\partial y\partial x}}\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55af531a2b1d239d67433ec34c7a2b94443965)
and ![{\displaystyle {\frac {\partial ^{2}\phi \ }{{\partial x}{\partial y}}}=(\phi \ _{x})_{y}\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/13c8b8438e29920fe9722e26ccc56ca57d860068)
and ![{\displaystyle {\frac {\partial ^{2}\phi \ }{{\partial y}{\partial x}}}=(\phi \ _{y})_{x}\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aea078d111cc618005f3e71319b02f842650469)
|
(1)
|
Application: Eq.(1)p.4-3 Not Exact
![{\displaystyle M(x,y)=2x^{2}+{\sqrt {y}}\Rightarrow \ M_{y}={\frac {1}{2{\sqrt {y}}}}\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1794969a1cc13b9e96c45822cf6e68aa5bc5c4eb)
![{\displaystyle N(x,y)=x^{5}y^{3}\Rightarrow \ N_{x}=5x^{4}y^{3}\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f3cc10934222f0f82634bd410cadd96f627d154)
![{\displaystyle M_{y}\neq \ N_{x}\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/afd7b23eebfa1187e8aaf85d56c8d1171fc342fd)