What is the interval for the particular solution?

In summary, the given equation has a particular solution of 3y³-11y = 3x³-3 and is defined for -⅓√11<y<⅓√11. To find the domain of x, one must find the range of the function of y, which is 3y³-11y over the interval of definition. The domain of x is then defined by the range of the function of y, restricted to the range found.
  • #1
faradayscat
57
8

Homework Statement


dy/dx = (9x²)/(9y²-11) , y(1)=0

Homework Equations

The Attempt at a Solution


I've found the particular solution:

3y³-11y = 3x³-3

The solution is defined for -⅓√11<y<⅓√11,
What about the x values? Plugging the y values into the equation doesn't work of course..
 
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  • #2
It looks to me that your solution is defined everywhere...how did you conclude the interval above?
 
  • #3
I see now...since you start with y=1, there is no way to continuously make it outside that interval -- where ##9y^2 - 11 = 0##.
To solve for the domain of x, I would recommend finding the range (max, min) of the function of y over the interval of definition. This would give you a domain defined by:
## 3x^3 - 3 \in \left[ \min_{y \in ( -\sqrt{11}/3, \sqrt{11}/3)} \{3y^3 -11\} , \max_{y \in ( -\sqrt{11}/3, \sqrt{11}/3)} \{3y^3 -11\}\right].##
 
  • #4
RUber said:
I see now...since you start with y=1, there is no way to continuously make it outside that interval -- where ##9y^2 - 11 = 0##.
To solve for the domain of x, I would recommend finding the range (max, min) of the function of y over the interval of definition. This would give you a domain defined by:
## 3x^3 - 3 \in \left[ \min_{y \in ( -\sqrt{11}/3, \sqrt{11}/3)} \{3y^3 -11\} , \max_{y \in ( -\sqrt{11}/3, \sqrt{11}/3)} \{3y^3 -11\}\right].##

Yes I understand all of this, however my assignment wants me to show the interval of existence like this:

A < x < B

That is, to find the domain, not the range. How can I do that?
 
  • #5
I don't think actual values are possible here? Only approximations of "x" for values greater by closest to -⅓√11 and smaller but closest to ⅓√11...
 
  • #6
The function of y, ##f(y) = 3y^3-11y## has its own domain and range. The domain is ##-\sqrt{11}/3 < y < \sqrt{11}/3. ## What is its range?
If ##g(x)=3x^3 - 3##, then the range of f(y) must equal the range of g(x). Then you find the domain of x that restricts g to the range you have found.
For example if the minimum of f(y) is -10, you would say:
## g(x) > -10 \\ 3x^3 - 3 > -10 \\ x^3 > -7/3 \\ x > (-7/3)^{1/3}##
 

What is the interval for the particular solution?

The interval for the particular solution refers to a range of values for the independent variable that satisfies the given equation or inequality.

Why is it important to determine the interval for the particular solution?

Determining the interval for the particular solution is important because it helps to identify all possible solutions to the given problem. It also helps to ensure that the solution is valid and does not violate any constraints.

How do you find the interval for the particular solution?

The interval for the particular solution can be found by setting the equation or inequality equal to zero and solving for the variable. The resulting solutions will determine the range of values for the independent variable.

Can the interval for the particular solution be negative?

Yes, the interval for the particular solution can be negative. It all depends on the given problem and the range of values that satisfy the equation or inequality.

What happens if the interval for the particular solution is infinite?

If the interval for the particular solution is infinite, it means that there is no limit to the range of values for the independent variable that satisfy the given equation or inequality. This could occur if the equation or inequality has no constraints or if the solution is a continuous function.

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