User:EGM6341.s11.TEAM1.WILKS/Mtg9

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010= Mtg 9: Thu, 9 Sep 10

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NOTE: Smooth functions are functions that are infinitely differentiable, e.g., $$ \cos x, e^x, \log x,x,x^2 \ $$ Non-Smooth:

The derivative is a generalized function: $$ \frac{dH (x - \hat x)}{dx} = \delta\ (x-\hat{x}) \ $$

Continuity of $$ f(x) \ $$ at $$ \hat{x} \ $$ : $$ \lim_{x \to \hat{x} ^-}f(x) = \lim_{x \to \hat{x} ^+}f(x) \ $$

Derivative in the generalized sense.

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Hat function not differentiable in classical sense at $$ x_1, x_2, x_3 \ $$ since $$ \lim_{x \to x_i ^-} \frac{df(x)}{dx} \ne \lim_{x \to x_i ^+} \frac{df(x)}{dx} \ $$ for $$ i=1,2,3 \ $$ limit on left of $$ x_i \ $$: $$ \lim_{x \to x_i ^-} \frac{df(x)}{dx} \ $$ limit on right of $$ x_i \ $$: $$ \lim_{x \to x_i ^+} \frac{df(x)}{dx} \ $$ smooth $$ \equiv \ $$ no kinks END NOTE HW  $$ M(x,y) + N(x,y) f(y')=0 \ $$, Find $$ f(y') \ $$ such that there is no analytical solution to $$ f(y')=-\frac{M}{N} \ $$ (i.e., such N1_ODE cannot be exact) END HW SUMMARY: 2 conditions of exactness 1) [[media: 2010_09_09_13_54_49.djvu | Eq.(3)P.8-2]] 2) [[media: 2010_09_09_13_54_49.djvu | Eq.(2)P.8-4]] END SUMMARY [[media: 2010_09_07_14_59_22.djvu | Ex. P.7-3]], [[media: 2010_09_09_13_54_49.djvu | P.8-2]]: [[media: 2010_09_09_13_54_49.djvu | Eq.(3)P.8-2]]

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Verify 2 conditions of exactness: 1) Condition 1: [[media: 2010_09_09_13_54_49.djvu | Eq.(3)P.8-2]] : where $$ M(x,y)=75x^4 \ $$ and $$ N(x,y)= \cos y \ $$ 2) Condition 2: $$ \Rightarrow \ \ $$ [[media: 2010_09_09_13_54_49.djvu | Eq.(3)P.8-2]] is exact. Suppose $$ \phi\ \ $$ can be found to be [[media: 2010_09_07_14_59_22.djvu | Eq.(4)P.7-3]], i.e., a reduced order equation $$ \phi\ (x,y)=k \ $$. If $$ y \ $$ can be solved in terms of $$ x \Rightarrow \ \ $$ explicit solution (otherwise get implicit solution) Here: Where $$ k \ $$ is the integration constant HW  verify Eq(4) satisfies Eq(1) END HW Question: Suppose condition 1 of exactness (continued on p9-4)

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(continued from 9-3) is satisfied, i.e. [[media: 2010_09_09_13_54_49.djvu | Eq.(3)P.8-2]] but not condition 2 of exactness, i.e., Can we make the original N1_ODE exact? $$ \Rightarrow \ \ $$ Euler integration factor method Question: Find $$ h(x,y) \ $$ such that Eq(2) is exact.