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I would like to solve for $x$ in the equations

$$a=e^{-x}(m(1+x+x^2/2+x^3/3)-x^3/3)$$

and

$$a=e^{-x}(m(1+x+x^2/2+x^3/3+x^4/4)-x^4/4)$$

or even with more terms. Here $a$ is some constant.

Does anyone have a hint?

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Do you mean that $x$ is to satisfy both of these equations (so simultaneous equations)? I would guess not, and in that case I would expect no simple closed form solutions, but only a numerical method for given $a,m$ could be applied. –  coffeemath Oct 28 '12 at 10:25

1 Answer 1

Even the simpler problem $a=e^{-x}(m(1+x)-x)$ (assuming $m\ne1$) can only be solved for $x$ by introducing the Lambert $W$-function, which you are invited to look up.

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yep, that's the simplest case, but with more terms, it gets harder to solve. –  chenwan Oct 29 '12 at 10:28

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