How to prove that the dependent variable could not be expressed explicitly in terms of the independent variable(s)? Consider the equation that xy=logy+1.

dalllc

dalllc

Answered question

2022-10-05

How to prove that the dependent variable could not be expressed explicitly in terms of the independent variable(s)?
Consider the equation that
x y = log y + 1 .
How does one prove that y cannot be expressed explicitly in terms of x?
By the way, I do not know how the adverb "explicitly" is strictly defined, though likely it is generally understood. For example, I would call y = x 1 sin 3 x + 1 an explicit expression.

Answer & Explanation

oldgaffer1b

oldgaffer1b

Beginner2022-10-06Added 9 answers

You can express y explicitly by x via the Lambert W function. You have
x y = log ( y ) + 1   e x y = y e   x e 1 = x y e x y   W ( x e 1 ) = x y   y = 1 x W ( x e 1 )

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?