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Subject: "Solving equations"

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Conferences The CTK Exchange High school Topic #324
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Reading Topic #324

Smith

guest

Jun-30-05, 12:11 PM (EST)

"Solving equations"

This is probably really simple but I cannot figure it out. Solve the following equation for x:

y = x + e^x + 3

To simplify, you must take the natural log since ln e^x = x, right?

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alexb
Charter Member
1572 posts

Jun-30-05, 12:13 PM (EST)

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1. "RE: Solving equations"
In response to message #0

>This is probably really simple but I cannot figure it out.
>Solve the following equation for x:
>
>y = x + e^x + 3
>

What do you mean by solve? For any x there is a value of y, so there are as many solutions as there are real numbers.

>To simplify, you must take the natural log

The equation is simple enough as it is. Taking logarithm will in no way simplify it.

< ln e^x = x,
> right?

This is correct.

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Smith

guest

Jul-01-05, 09:09 AM (EST)

2. "RE: Solving equations"
In response to message #1

No, not simplify...solve for x as if I was given a y value and needed to find the corresponding x value.

y = x + e^x +3

y - 3 = x + e^x

???

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alexb
Charter Member
1572 posts

Jul-01-05, 09:14 AM (EST)

3. "RE: Solving equations"
In response to message #2

>No, not simplify...solve for x as if I was given a y value
>and needed to find the corresponding x value.
>
>y = x + e^x +3
>
>y - 3 = x + e^x

The function f(x) = x + e^x is monotone. Thus, in principle, for every y there is a unique x. For some y, e.g., y = 4, you can easily guess the corresponding x. E.g., y(0) = 4. Other than that, the equation is not solvable in closed form using ln or other elementary functions. You may want to consider approximate solutions.

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Smith

guest

Jul-01-05, 12:44 PM (EST)

4. "RE: Solving equations"
In response to message #3

Okay, thank you. This was given to me by a calculus student who says it was a textbook problem that does not show the answer. Sorry, I don't have a reference for the text.

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guest

Jul-02-05, 08:26 AM (EST)

5. "RE: Solving equations"
In response to message #0

The roots of the equation :
y = x + e^x + 3
cannot be expressed in terms of elementary functions.
But whey can be expressed in terms of infinite series.
The equation can be analytically solved, thanks to the Lambert's function W(X). The result is :
x = y-3 -W(X) where X = exp(y-3)
Bassic information on the LambertW function :
https://mathworld.wolfram.com/LambertW-Function.html
More information on infinite series development and computation :
https://les-mathematiques.u-strasbg.fr/phorum/read.php?f=2&i=96315&t=96315

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