- 1 R4.1 Direct Derivation of Alternate Version of RTT
- 2 R*4.2 Exactness Verification
- 3 Problem R*4.3: Find and of the integrating factor such that the given N2-ODE is exact then solve for .
- 4 R*4.4 Generate a class of exact L2-ODE-VC
- 5 R*4.5 Solve a L2-ODE-VC
- 6 R*4.6 Show the equivalence of two forms of 2nd exactness condition of N2-ODE
- 7 References
- 8 Team Work Distribution
R4.1 Direct Derivation of Alternate Version of RTT
Another version of the Reynolds Transport Theorem:
Provide a different and direct derivation of Equation 4.1.1.
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Solution is adapted from Malvern. If denotes any property of the material volume , for a spatial volume bounded by a control surface , the following is true:
Now expressing mathematically what is stated above:
R*4.2 Exactness Verification
Example: Consider the following ODE:
Verify the exactness of Equation 4.2.1
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To be exact, the equation 4.2.1 must satisfy two conditions.
1st Condition of Exactness
Fist, the equation must be presented in the following format
By letting y'=p, the first condition of exactness is satisfied as shown in the following equation:
2nd Condition of Exactness
In order to meet the second condition of exactness, the following equations must be satisfied:
Each element of equationgs 4.2.4 and 4.2.5 are calculated to be the following:
Plugging each relevant element into equation 4.2.4 yields the following:
Plugging each relevant element into equation 4.2.5 yields the following:
Equations 4.2.4 and 4.2.5 have not been satisfied, therefore proved equation 4.2.1 is not exact.
Problem R*4.3: Find and of the integrating factor such that the given N2-ODE is exact then solve for .
A function is given by
Per Lecture notes: Mtg 21 (c) where
Part A: Find such that is exact.
Part B: Show that the first integral is a L1-ODE-VC and solve
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The first exactness criteria for second order non-liner ODE is
Comparing (1) p.21-3 with the form of (2) p.16-4, the functions and can be defined as
So the form of the 1st exactness condition is met.
For a second order non-liner ODE, the first part of the second exactness condition is
For a second order non-liner ODE, the second part of the second exactness condition is
Using the second part of the second exactness condition given above in and recognizing the given from equation is not a function of , so the first two terms are zero. Additionally is constant wrt. and therefore is zero. can be simplified as shown below
Taking the derivative .wrt of in equation and substituting into yields
In order for equation to be true
For clarity we can now re-write equations with as shown below:
Next, use the first part of the the second exactness condition given in equation with the following derivatives of and plugged in.
Solving this equation for yields
So the integrating factor to make exact is
So using the integrating factor found above and multiplying through to equation to get and exact N2-ODE one can use equation to obtain the first integral given in
Rearranging equation and solving for by dividing through by x to put the ODE into the general form:
The ODE can be solved by finding an integrating factor per
Substituting in gives:
Integrating which is given in
Plugging in and into yields:
Equation is the solution for
R*4.4 Generate a class of exact L2-ODE-VC
Question from meeting 21, pages 5-6:
Given (2)-(3) from p. 21-4
Find that the solution gives (1) from p. 21-5
From lecture notes Mtg 21
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The equations (2) and (3) are within the category of L2-ODE-VC. The key is to use the substitution p = y' for the derivations and p' = y.The first step is to prove the exactness of the equations below. There are two exactness conditions for the N2-ODE. Note that the functions R and Q are of x but not y. This affects the 2nd exactness condition and subsequent derivations and integrations.
The function with p=y' can be rearranged.
The equations satisfy the following form of the 1st exactness condition.
The 2nd exactness condition is derived below. First, assume that
Then the 2nd exactness condition can be applied satisfactorily.
This can be shown by placing the values of R and Q into the equation for 2nd exactness.
The next step is to find the integration factor h(x) from the equation below.
Recall that the integration factor in this case is a function of x, so the following equation from meeting 11 can be used.
The integration is simple.
Subsequently, the final solution is equated below.
References for R4.4Vu-Quoc, L. Class Lecture: Principles of Engineering Analysis. University of Florida, Gainesville, FL, (Meeting 11) Mtg 11 13 Sep 2011.
Vu-Quoc, L. Class Lecture: Principles of Engineering Analysis. University of Florida, Gainesville, FL, (Meeting 16) Mtg 16 22 Sep 2011.
Vu-Quoc, L. Class Lecture: Principles of Engineering Analysis. University of Florida, Gainesville, FL, (Meeting 21) Mtg 21 04 Oct 2011.
R*4.5 Solve a L2-ODE-VC
1）Show Equation(4.5.1) is exact.
3）Solve for .
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The particular form of N2-ODE-VC is
This is shown that Equation(4,5,1) satisfies the 1st exactness condition.
To verify its exactness, we calculate following terms:
Recall the 2nd exactness condition for L2-ODE-VC:
Substituting those calculated terms into Equation(4,5,2) and Equation(4.5.3).
As for Equation(4,5,2) ,
As for Equation(4,5,3) ,
Therefore, Equation(4,5,1) is exact.
We find the first integral
Substituting those partial derivatives of into function g.
To solving h(x,y), we assume that
Partial derivate respect to y,
We can find that
Therefore, the first integral is
we obtain a L1-ODE-VC that
Rewrite it into
Therefore, the integration factor is
The solution is
R*4.6 Show the equivalence of two forms of 2nd exactness condition of N2-ODE
It's equal to the 2nd exactness condition in form:
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For we have:
So, both two parts in the left side of equation 4.6.1 are equal to 0 separately:
For first part:
And, second part:
thus we get:
- Vu-Quoc, L. Class Lecture: Principles of Engineering Analysis. University of Florida, Gainesville, FL Meeting 19-20 29 Sep 2011
- Malvern, Lawrence. Introduction to the Mechanics of a Continuous Medium. pgs 210-211. New Jersey: Prentice-Hall, 1969. Print.
- Vu-Quoc, L. Class Lecture: Principles of Engineering Analysis. University of Florida, Gainesville, FL Meeting 21 4 Oct 2011
- Vu-Quoc, L. Class Lecture: Principles of Engineering Analysis. University of Florida, Gainesville, FL Mtg 22 19 Oct 2011
Team Work Distribution
Problem Assignments Problem # Assigned To R4.1 Fenner Colson R*4.2 Yi Zhao R*4.3 Ben Neri R*4.4 Manuel Steele R*4.5 Jing Pan R*4.6 Zexi Zheng Table of Contributions Name Problems Solved Problems Checked Signature Fenner Colson R4.1 R*4.5 Egm6321.f11.team1.colsonfe 03:13, 19 October 2011 (UTC) Yi Zhao R*4.2 Egm6321.f11.team1.yizhao Manuel Steele R*4.4 R*4.3 Egm6321.f11.team1.steele.m 07:16, 19 October 2011 (UTC) Jing Pan R*4.5 R4.1 R*4.2 Egm6321.f11.team1.pan 15:52, 19 October 2011 (UTC) Benjamin Neri R*4.3 R*4.2 Egm6321.f11.team1.Benneri Zexi Zheng R*4.6 R*4.4 Egm6321.f11.team1.zheng.zx