plotting - Adding a gradient filling according to a given function between two curves

Consider the following sample code

f1[x_]:=x;
f2[x_]:=2x;
 c[y_]:= Exp[-y];  

Plot[{f1[x], f2[x]}, {x, 0, 1}, 
     PlotStyle -> {ColorData["SolarColors", 0], ColorData["SolarColors", 1]}]

Plot[c[2 x - x], {x, 0, 1}]

I would like to have a filling that fills the area between the two curves f1[x],f2[x] by the values of c[f2[x]-f1[x]] in the following manner

1. That the values of c[f1[x]] and c[f2[x]] will be the end values of the colordata.


2. The area in between them will be a gradient fill with the colordata that is normalized according to c[f2[x]-f1[x]]

What I want is best presented in the figure

Answer

Is this what you are looking for?

Plot[{f1[x], f2[x]}, {x, 0, 1}, Filling -> {1 -> {2}}, 

     ColorFunction -> Function[{x, y}, 
     Blend[{ColorData["SolarColors", 0], ColorData["SolarColors", 1]}, 
     c[(y - f2[x])/(f1[x] - f2[x])]]],
     ColorFunctionScaling -> False
]

It looks better if you normalise $\exp$ to the range $[0,1]$ instead of $[0,e^{-1}]\approx [0,.4]$:

c[y_] := (Exp[-y] - Exp[-1])/(1 - Exp[-1])