How Can You Solve These Variable Equations?

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In summary, the conversation discusses solving for x and y in equations involving the Lambert W function and exponentials. The last equation, solving for y, does not have a closed-form solution and another unsolvable problem is also mentioned.
  • #1
FizX
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How would you go about solving these?

solving for x, x ln(x)=A

solving for y, y Exp[y] =B

and also what would happen if you turned A into A+x and B into B+y.

Thanks
 
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  • #2
Use Lambert's W function which is DEFINED as the inverse function to f(x)= xex.
 
  • #3
[tex] x \ln(x) = A \Rightarrow x = \frac{A}{W\{A\}}[/tex]
[tex] ye^{y} = B \Rightarrow y = W\{B\}[/tex]
[tex] x \ln(x) = A + x \Rightarrow x = \frac{A}{W\{\frac{A}{e}\}}[/tex]
[tex] y e^{x} = A + y \Rightarrow y = ? [/tex]

I am still working on the fourth one...looks hard!

edit: sorry, the last one is suppose to have a y in the argument of the exponential instead of an x.
 
Last edited:
  • #4
I don't know! Someone else, help!
 
  • #5
Swapnil said:
I don't know! Someone else, help!

Can you factor out the y?
 
  • #6
Leon22's logic is right, subtract y from the righthand side and place it with the terms y*e^x to get an expression that looks like:

y*(e^(x))-y=A

You can then factor out the y, and get an expression:

y* ((e^(x))-1)=A.

Then just solve for y.

Sorry I don't have much experance with the function that was referanced eariler and didn't bother trying to put into that format, but the solution gained from leon22's logic should be vaild.

Good Luck.
 
  • #7
Oops... There wasn't suppose to be any x in that equation. All the variable were suppose to be y. That's why I had to resort to using the Lambert W funcition. It was suppose to be:

[tex] y e^{y} = A + y \Rightarrow y = ? [/tex]

Sorry about that. So does anyone now know how to solve for y?
 
  • #8
I tried mathematica too but no luck. I am guessing that it is probably not possible to solve for y in closed-form.

BTW, here is another problem that I came across a while back which doesn't seem to have a closed-form solution either:

[tex]x^x = Ax[/tex]
 
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1. What is the purpose of solving for a variable?

Solving for a variable allows us to find the value of an unknown quantity in an equation or problem. It helps us to understand and manipulate mathematical relationships.

2. How do you solve for a variable?

To solve for a variable, you need to isolate the variable on one side of the equation using basic algebraic operations such as addition, subtraction, multiplication, and division. The goal is to get the variable by itself on one side of the equals sign.

3. What are the common methods used to solve for a variable?

The most common methods used to solve for a variable are the addition and subtraction method, the substitution method, and the elimination method. These methods involve manipulating equations to isolate the variable and solve for its value.

4. What do you do if there are multiple variables in an equation?

If there are multiple variables in an equation, you can still solve for one variable by treating the other variables as constants. This means that their values will not change during the solving process and you can still isolate the variable you are solving for.

5. Can you solve for a variable if there are no numbers in the equation?

Yes, it is possible to solve for a variable even if there are no numbers in the equation. This is because the same algebraic principles apply regardless of whether there are numbers or not. You can still use the methods mentioned above to isolate the variable and solve for its value.

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