Solving Equations with the Lambert W Function: A* (e^2x - e^x) + b*x = c

[jya]

I know we can solve e^x=x by the Lambert W function, but is it possible to solve the following equation:

a*(e^(2x)-e^x)+b*x=c

in terms of a, b, and c.

 

[Simon Bridge]

Welcome to PF;

solve: a*(e^(2x)-e^x)+b*x=c ... in terms of a, b, and c.

Solving for x given a,b,c you mean? That would be the intersection of a line with a quadratic in e^x... that is $$ e^{x}\left ( e^x - 1\right ) = mx+k$$ ...where $m=-b/a$ and $k=c/a$.
And you want to find x given m and k.
That help?

 

[jya]

Thank you Simon. Yes. I am wondering if there is some special function can solve this problem, i.e. given the values of a,b,c (or m,k in your equation) I can find out the value of x without looking at the graph.

Welcome to PF;

Solving for x given a,b,c you mean? That would be the intersection of a line with a quadratic in e^x... that is $$ e^{x}\left ( e^x - 1\right ) = mx+k$$ ...where $m=-b/a$ and $k=c/a$.
And you want to find x given m and k.
That help?