How do I solve y*exp(y)=[constant] for J using the Lambert W function?

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In summary, the conversation was about trying to solve for J in an equation that had both J terms on the left and right sides. The simplified terms were J01=1, q/kt=1, Rs=1, Rshunt=infinity, V=0. The solution to this problem is the Lambert W function, which cannot be expressed in terms of elementary functions. There are papers on using this function for the double-diode problem. Wolfram Alpha gives the solution to y*exp(y)=5 as y=W(5). The person asking for help was grateful for the information and was glad to learn how to solve the equation.
  • #1
willDavidson
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TL;DR Summary
I would like to solve an equation that ends up in a form similar to y*exp(y)=[some constant]
I am trying to solve for J in the equation attached. I've also posted a link to the equation. The equation is to solve for J but both the left and right sides have J. I'm not quite sure how to solve that and it looks really messy so I simplified. Letting all of the terms equal either 0 of 1, I get J*exp(J)=2. The terms I simplified are below.
Let J01=1, q/kt=1, Rs=1, Rshunt=infinity, V=0

I'm stuck on what to do with this to solve for J so any help is appreciated.

https://www.pveducation.org/pvcdrom/characterisation/double-diode-model
 

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The solution is the Lambert W function, which cannot be expressed in terms of elementary functions:

https://en.wikipedia.org/wiki/Lambert_W_function

There are also papers on using Lambert W for the solution of the double-diode problem, but I don't know anything about that topic, so I will not link any here.

By the way, I just checked on wolfram alpha, and it gives y=W(5) as the solution to y*exp(y)=5
 
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  • #3
bigfooted said:
The solution is the Lambert W function, which cannot be expressed in terms of elementary functions:

https://en.wikipedia.org/wiki/Lambert_W_function

There are also papers on using Lambert W for the solution of the double-diode problem, but I don't know anything about that topic, so I will not link any here.

By the way, I just checked on wolfram alpha, and it gives y=W(5) as the solution to y*exp(y)=5

Ahh thanks. I really wanted to know how to actually solve this thing so this will help a lot.
 

FAQ: How do I solve y*exp(y)=[constant] for J using the Lambert W function?

How do I solve the equation y*exp(y)=[constant]?

The first step in solving this equation is to rewrite it in the form of a natural logarithm: y*exp(y)=[constant] becomes y + ln(y) = ln([constant]). From there, you can use algebraic manipulation and logarithm rules to solve for y.

Can this equation be solved analytically or do I need to use numerical methods?

This equation can be solved analytically using algebraic manipulation and logarithm rules. However, if the constant is a complex number or if the equation is part of a larger system of equations, numerical methods may be necessary.

What is the significance of the exponential function in this equation?

The exponential function is significant in this equation because it is what makes the equation non-linear. Without the exponential function, the equation would be a simple linear equation that is much easier to solve.

Are there any special cases or restrictions for solving this equation?

One special case to consider is when the constant is equal to 0. In this case, the equation becomes y*exp(y) = 0, which has two solutions: y=0 and y=-∞. Additionally, the solution for y may be restricted to certain values depending on the context of the problem.

How can I use the solution to this equation in real-world applications?

The solution to this equation can be used in a variety of fields, including physics, chemistry, and economics. It can be used to model exponential growth or decay, as well as to solve problems involving rates of change. It can also be used to find optimal solutions in optimization problems.

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