Solving Difficult Problem: $\partial x/\partial y$ as Function of x

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In summary, the conversation discusses solving for y as a function of x using an incomplete elliptic integral. The difficulty of this problem is acknowledged, but a method for solving it is mentioned and a reference is promised for further information. The complexity of mathematics is also mentioned, with the example of understanding integers in cyclotomic fields as an example of a difficult problem.
  • #1
redtree
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Given the following:

[tex]\partial x[/tex]/[tex]\partial y[/tex] = A * [B/y + C/(y-D)]^(1/2)

Solve for y as a function of x.
 
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  • #2
Replacing the partial of x with respect to y with dx/dy, and moving the dy from the denominator of the l.h.s. to the numerator of the r.h.s., and then taking the indefinite integral of both sides, does not that express x in terms of an (incomplete) elliptic integral of y?
 
  • #3
DeaconJohn said:
Replacing the partial of x with respect to y with dx/dy, and moving the dy from the denominator of the l.h.s. to the numerator of the r.h.s., and then taking the indefinite integral of both sides, does not that express x in terms of an (incomplete) elliptic integral of y?

Solving for x as a function of y is easy. It's just as you suggested. The hard part is solving for y as a function of x (at least for me).
 
  • #4
Well, if it is an elliptic integral like I suspect, then that is a classic and difficult problem. Many very good mathematicians were unable to make any progress on it for many years. However, there is now a known method of solving it. That is, if I remember correctly. In any case, it's the kind of stuff Ramanujan was good at. I post a reference next time I run across one.

Nobody said mathematics was easy. Many elementary things are beyond our current grasp. Mathematicians don't even understand which integers in cyclotomic fields are units. It's because there are a lot of hard problems.
 
  • #5
DeaconJohn said:
Well, if it is an elliptic integral like I suspect, then that is a classic and difficult problem. Many very good mathematicians were unable to make any progress on it for many years. However, there is now a known method of solving it. That is, if I remember correctly. In any case, it's the kind of stuff Ramanujan was good at. I post a reference next time I run across one.

Nobody said mathematics was easy. Many elementary things are beyond our current grasp. Mathematicians don't even understand which integers in cyclotomic fields are units. It's because there are a lot of hard problems.

Thanks. I look forward to it.
 

1. What is the purpose of solving a difficult problem involving $\partial x/\partial y$ as a function of x?

The purpose of solving a difficult problem involving $\partial x/\partial y$ as a function of x is to find the relationship between two variables, x and y, and to determine how changes in x affect the value of $\partial x/\partial y$.

2. What is the difference between $\partial x/\partial y$ and $\frac{\partial x}{\partial y}$?

The notation $\partial x/\partial y$ represents the partial derivative of x with respect to y, while $\frac{\partial x}{\partial y}$ represents the same derivative but in fraction form. Both notations are used interchangeably and mean the same thing.

3. How do I solve a difficult problem involving $\partial x/\partial y$ as a function of x?

To solve this type of problem, you will need to use the chain rule and implicit differentiation. You will also need to set up an equation with x and y as variables and then take the partial derivative of both sides with respect to x.

4. What are some real-world applications of solving problems involving $\partial x/\partial y$ as a function of x?

This type of problem can be applied in various fields such as physics, economics, and engineering. For example, it can be used to determine the relationship between temperature and pressure in a gas, or the relationship between demand and price in economics.

5. Can I solve a problem involving $\partial x/\partial y$ as a function of x without using calculus?

No, this type of problem requires the use of calculus, specifically the concepts of derivatives and partial derivatives. It is not possible to solve these types of problems without using these mathematical tools.

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