How to solve exponential equation $a + bx + c\exp(x) + dx\exp(x) = 0$?

Question

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!

Answer 1

$$a + bx + c\exp(x) + dx\exp(x) = 0 \implies e^{-x}=-\frac{c+dx } {a+bx }$$ 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.