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I'm trying to solve:

$a + bx + c\exp(x) + dx\exp(x) = 0$

I didn't get any further than rewriting it as:

$(x+c/d)(\exp(x)+b/d) = bc - a/d$

Which doesn't get me anywhere. If someone is able to help me with this that would be great!

31Noah
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    I'm pretty sure this has no closed-form solution. And it may have anywhere from zero to one to two solutions, depending on the values of the constants. Can you expand more on the context here? – Ben W Mar 20 '20 at 03:55
  • Thank you for the quick response. The reason I write it in this general form is that the values of a, b, c and d are complex (a function of several parameters) and irrelevant. I need to solve this equation as part of a research project. – 31Noah Mar 20 '20 at 21:57

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$$a + bx + c\exp(x) + dx\exp(x) = 0 \implies e^{-x}=-\frac{c+dx } {a+bx }$$ So, the equation can be solved in terms of the generalized Lambert function (have a look at equation $(4)$).

This being said, think about a numerical method such as Newton.

Claude Leibovici
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