How Does the Solution of y' = 1/(1+x^2+y^2) Behave as x Approaches Infinity?

  • Thread starter Zaare
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In summary, limes are commonly used to determine the pH level of an unknown solution due to their citric acid content. Limes contain a natural indicator called anthocyanin that changes color depending on the pH level of the solution. They can be used for most types of solutions, but may not be as effective for extreme pH levels. The amount of lime to be used depends on the volume of the solution and desired strength of the indicator. Safety precautions should be taken when handling limes and the solution to avoid skin and eye irritation, and to ensure accurate results.
  • #1
Zaare
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I have this eq.:
[tex]y'=\frac{1}{(1+x^2+y^2)}[/tex]
I'm able to show that it has a unique solution for [tex]-\infty<x<\infty[/tex], and that the solution is an odd funktion.
What can I say about the limit of the solution as x grows towards infinity?
 
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  • #2
For ALL x, you have:
[tex]0\leq{y'}\leq\frac{1}{1+x^{2}}[/tex]

How does this help you?
 
  • #3
Right! That means, [tex]y\leq{\arctan{x}}[/tex], therefor the limit of y is [tex]\leq{\pi/2}[/tex].
Thanks. I appreciate the help. :)
 

Related to How Does the Solution of y' = 1/(1+x^2+y^2) Behave as x Approaches Infinity?

1. What is the purpose of using limes for an unknown solution?

The use of limes in an unknown solution is to determine the pH level of the solution. Limes contain citric acid, which is a weak acid that can react with the solution and change color depending on the pH level.

2. How do limes indicate the pH level of an unknown solution?

Limes contain a natural indicator called anthocyanin that changes color depending on the pH level of the solution. When added to an unknown solution, limes will change color to indicate whether the solution is acidic, basic, or neutral.

3. Can limes be used for all types of unknown solutions?

Limes can be used for most types of unknown solutions, but their effectiveness may vary depending on the chemical composition of the solution. It is best to use limes for solutions that are slightly acidic or basic, as extreme pH levels may not produce accurate results.

4. How much lime should be used for an unknown solution?

The amount of lime to be used for an unknown solution will depend on the volume of the solution and the desired strength of the indicator. Generally, 1-2 drops of lime juice per 10mL of solution should be sufficient to indicate the pH level.

5. Are there any safety precautions to consider when using limes for an unknown solution?

Yes, it is important to wear gloves and goggles when handling limes, as the citric acid can irritate the skin and eyes. Additionally, be careful when handling the solution as the color change may not be immediate and can lead to inaccurate results if not given enough time to fully react.

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