Tenenbaum & Pollard Pages 53-54: Apparently does not make sense

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In summary, the conversation is discussing a differential equation and its solution, specifically when the derivative is infinite and when y is less than 1. The conversation also covers the confusion about the graph of the solution not being asymptotic to the vertical line x=1. It is clarified that the illustration is erroneous but the solution is correct. The conversation ends with a solution and Matlab code for visual representation.
  • #1
xorg
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Hi. I'm reading the bookhttp://amzn.com/0486649407 ,
S7ShG10.jpg

in self-study mode.
In page 53 and 54, below:
voXXhL8.jpg

Apparently does not make sense, because, If the differential equation is:
$$ \frac{\mathrm{d}y }{\mathrm{d} x} = x\frac{\sqrt{1-y}}{\sqrt{1-x^{2}}} $$

then dy/dx = ∞ when x = 1, and y < 0
however, in Image:
Ifx1MI0.png

dy / dx is not ∞ in point marked in green.

Then, in this point, the graph changes radically the slope? Apparent, does not make sense, It seems that the correct curve would go up until tangent to x = 1, while the particular solution says otherwise.

Why the differential equation "fails", and the solution of the differential equation not? I did not catch that.

I am studying alone, thank you for your patience.
 
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  • #2
There is some gap in your knowledge about what it means for a graph to have a derivative. I would advice you read a calculus book and brush up on it. This idea is a very fundamental key point in ode.
 
  • #3
To clarify the question: Why does the graph show a solution curve that is not asymptotic to the vertical line x = 1 ?

xorg said:
Apparently does not make sense, because, If the differential equation is:
$$ \frac{\mathrm{d}y }{\mathrm{d} x} = x\frac{\sqrt{1-y}}{\sqrt{1-x^{2}}} $$

then dy/dx = ∞ when x = 1, and y < 0

Did you mean "when [itex] y < 1 [/itex] ?

however, in Image:
Ifx1MI0.png

dy / dx is not ∞ in point marked in green.
 
  • #4
Stephen Tashi said:
To clarify the question: Why does the graph show a solution curve that is not asymptotic to the vertical line x = 1 ?

Yes, if the dy/dx go to ∞ when x go to 1, then i guess in particular solution this should be the trend.
It is as if the particular solution and differential equation were conflicting.
Imagine that someone will draw the solution curves, from the dy / dx, at each point, does not match the presented solution.
However, the method of solution seems correct, that leaves me confused!

Did you mean "when [itex] y < 1 [/itex] ?

YES, when y<1. Sorry.
 
  • #5
xorg said:
Yes, if the dy/dx go to ∞ when x go to 1, then i guess in particular solution this should be the trend.
It is as if the particular solution and differential equation were conflicting.
Imagine that someone will draw the solution curves, from the dy / dx, at each point, does not match the presented solution.
However, the method of solution seems correct, that leaves me confused!
YES, when y<1. Sorry.

You are correct: the illustration is erroneous. The solution is correct, however. If you plot the family of solutions, you will get curves with vertical tangent lines x = 1 and x = -1 when y < 1, as described by the text.
In particular, if you isolate y in the family of solutions, you will get [itex]y = \frac{1}{4}x^2 + C\sqrt{1-x^2} + \frac{3}{4} - C^2[/itex], from which you can find the explicit derivative. You can see that it is only when C = 0 that the problematic derivative of the square root at x = 1, and at x = -1, disappears.
 

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  • #6
slider142 said:
You are correct: the illustration is erroneous. The solution is correct, however. If you plot the family of solutions, you will get curves with vertical tangent lines x = 1 and x = -1 when y < 1, as described by the text.
In particular, if you isolate y in the family of solutions, you will get [itex]y = \frac{1}{4}x^2 + C\sqrt{1-x^2} + \frac{3}{4} - C^2[/itex], from which you can find the explicit derivative. You can see that it is only when C = 0 that the problematic derivative of the square root at x = 1, and at x = -1, disappears.

Thank You!

The solution:
$$ \sqrt{1-x^{2}} - 2\sqrt{1-y} = C $$
Results:
$$ y = \frac{3}{4} + \frac{x^{2}}{4} + \frac{C\sqrt{1-x^{2}}}{2} - \frac{C^{2}}{4} $$
(in your solution, yourC = C/2)

Matlab Code
PHP:
x = linspace(-1,1,1000);
clf;
grid on;
hold on;
c=0;
y=3/4+(x.^2)/4 +(c/2)*sqrt(1-(x.^2))-(c/2)^2;plot(x, y, 'r');
c=-0.5;
y=3/4+(x.^2)/4 +(c/2)*sqrt(1-(x.^2))-(c/2)^2;plot(x, y, 'b');
c=-1;
y=3/4+(x.^2)/4 +(c/2)*sqrt(1-(x.^2))-(c/2)^2;plot(x, y, 'g');

Matlab Result
CqzeJeY.png


c = 0

c = -0.5
c = -1



Thank you, Helped me a lot!
 

1. What is the significance of "Tenenbaum & Pollard Pages 53-54"?

"Tenenbaum & Pollard Pages 53-54" is a reference to a specific section in a scientific paper written by Tenenbaum and Pollard. This section is likely discussing a particular topic or experiment related to the overall research being presented in the paper.

2. Why does the statement "Apparently does not make sense" appear in the title of the section?

The phrase "Apparently does not make sense" may indicate that the authors found some aspect of their research to be confusing or conflicting. It could also suggest that there are still unanswered questions or inconsistencies in the data presented.

3. What does it mean if a section of a scientific paper "does not make sense"?

If a section of a scientific paper "does not make sense," it could mean that the authors have encountered unexpected results or that there are discrepancies in their findings. It could also indicate that there are limitations or gaps in their understanding of the subject matter.

4. How does the statement "Apparently does not make sense" affect the overall credibility of the scientific paper?

The statement "Apparently does not make sense" does not necessarily affect the overall credibility of the scientific paper. It is common for researchers to encounter unexpected or confusing results in their studies. However, it is important for the authors to acknowledge and address these discrepancies in order to maintain credibility.

5. Can readers still trust the information presented in the "Tenenbaum & Pollard Pages 53-54" section?

As a scientist, it is important to critically evaluate all information presented in a scientific paper, including the "Tenenbaum & Pollard Pages 53-54" section. While the authors may have encountered some inconsistencies, this does not necessarily mean that the information presented is inaccurate. It is up to the reader to carefully consider all evidence and draw their own conclusions.

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