Square-root and differential equations

In summary, the conversation discusses solving a differential equation with a square root, and the restrictions that need to be considered in order for the solution to be valid. The conversation also briefly touches on the concept of open and closed sets in mathematics.
  • #1
Niles
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0

Homework Statement


Hi all

I have the following expression

[tex]
\frac{dy}{dt}= \sqrt{C-y(t)^2},
[/tex]

where C is a constant, and y is the variable, which depends on t. What I need to do is to solve this differential equation, but my problem is that this is a math-class (and not a physics-class), so I need to be very rigorous.

Now it is almost mandatory to comment of the argument of a squareroot. What I know is that the solution y is real.

Thus the term C-y(t) cannot be less than zero. But my question is: Does it make sense to talk about the term "C-y(t)" being less than zero, when y(t) is time-dependent?
 
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  • #2
Sure, you can insist [itex]y^2(t) \le C[/itex] or [itex]|y(t)| \le \sqrt{C}[/itex] for all t. Obviously C will have to be positive. If you just go ahead and solve it by separation of variables you will find that restriction automatically holds for your solution.
 
  • #3
LCKurtz said:
Sure, you can insist [itex]y^2(t) \le C[/itex] or [itex]|y(t)| \le \sqrt{C}[/itex] for all t. Obviously C will have to be positive. If you just go ahead and solve it by separation of variables you will find that restriction automatically holds for your solution.

Thanks. I must admit, I am not entirely sure what you mean when you say "restriction automatically holds for your solution.".

I have a final question, which is in no way related to this, but perhaps you can answer it anyway. Do you know if the set R2 (i.e. all of R2) can be considered as an open set?
 
  • #4
Niles said:
Thanks. I must admit, I am not entirely sure what you mean when you say "restriction automatically holds for your solution.".
Just go ahead and solve the DE. The y(t) that you get will work under the square root sign with no problems about a negative under the square root.
I have a final question, which is in no way related to this, but perhaps you can answer it anyway. Do you know if the set R2 (i.e. all of R2) can be considered as an open set?

Yes. R2 is open. And closed.
 
  • #5
Thanks! That was very kind of you.
 

FAQ: Square-root and differential equations

1. What is a square root?

A square root is a mathematical operation that determines the number that, when multiplied by itself, results in a given number. For example, the square root of 25 is 5 because 5 multiplied by itself equals 25.

2. How do you solve a square root equation?

To solve a square root equation, you need to isolate the square root term on one side of the equation and square both sides. For example, to solve the equation √x = 3, you would square both sides to get x = 9.

3. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It involves the use of differential calculus to find the rate of change of a variable over time or space.

4. How do you solve a differential equation?

Solving a differential equation involves finding a function that satisfies the equation. This can be done analytically or numerically using various methods such as separation of variables, substitution, or using software programs.

5. What are the applications of square-root and differential equations?

Square-root and differential equations are used in many scientific fields, including physics, engineering, economics, and biology. They are particularly useful in modeling and analyzing complex systems and phenomena that involve rates of change, such as population growth, chemical reactions, and motion.

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