# Inverse of an equation with x*e^x

Posted 6 months ago
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 I am trying to get the inverse (solve for x) of an equation with the form:y(x) = a * (1 - e^ (-x/b) - (x/b) * e^ (-x/b))x(y) = ?Through some research, I learned that the inverse of x*e^x is a Lambert function. However, when I use Wolfram|Alpha to perform the inverse, the solution does not reproduce the results of the original equation. WA returns many conditional solutions depending on values of a, b, and the range of y values. In addition, there are different branches of the Lambert function (Wn). When I use the branch n = -1, the solution is close, but not very good near the initial values. When I plot the solution using the main branch of W (n=0), the solution gives negative values. When I plot the real parts of the -1 branch, the solution is close in the middle and final values, but it is not fit well near the initial values. Any thoughts?
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Posted 6 months ago
 You may want to have a look at this fy[x_, a_, b_, n_] := a (1 - Exp[-x/b] - x/b Exp[-x/b]) fx[y_, a_, b_, n_] := -b - b ProductLog[n, (-a + y)/(a E)] fx[fy[.5, .1, .5, -1], .1, .5, -1] // FullSimplify fy[fx[.5, .1, .5, -1], .1, .5, -1] // FullSimplify (fx[fy[#, .1, .5, -1], .1, .5, -1] // FullSimplify) & /@ Range[20/2] (fy[fx[#, .1, .5, -1], .1, .5, -1] // FullSimplify) & /@ Range[20/2] // Chop 
 and also p1 = Plot[fy[x, .2, 1.3, n], {x, 0, 5}, PlotStyle -> {Thick, Dashed, Red}] p2 = ParametricPlot[{fx[y, .2, 1.3, -1], y}, {y, 0, .2}, AspectRatio -> .5] Show[p1, p2]