# Exponential Equation Solution

I have the following equation:

$$e^{-x}(1+x) = 0.935$$

How can I solve for $x$ by hand? I remember I learned an easy way to solve it, which I forgot. Any help?

Thanks,

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Your equation is NOT solvable by any direct means. The solution can be expressed with the so called, Lambert-W function. Here is a numerical solution

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As Alex R explains, this is solvable by application of the Lambert-W function:

So we have $e^{-x}(1 + x) = y$

Let $u = -(x + 1)$, then

$e^{1+u}(-u) = y$

Or equivalently

$e^1 e^u (-u) = y$

Meaning that $ue^u = -\frac{y}{e}$

Then Lambert-W gives $u = W(-\frac{y}{e})$

Reversing our substitution we find that

$x = -W(-\frac{y}{e}) - 1$

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