Consider an implicit function, $F(e,w,a,b,i,n)=0$, whose code is:
D[w - 1/((1 - e) w), e] == D[1 - e, e] ((-((-a (i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) + (-1 + a^2) (-1 + e)^2 w - a (-1 + e) (i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) w^2)/(a (-1 + e) i (1 - e + i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) w))) - (-((-a (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - n e (1/(b (a e)^(b)))^(1/(b - 1))) + (-1 + a^2) (-1 + e) (-1 + e - i) w - a (-1 + e) (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(
1 - n e (1/(b (a e)^(b)))^(1/(b - 1))) w^2)/( a (-1 + e) i (1 - e + i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) w))))
From the above implicit function, $e$ is implicitly defined as a function of $w$, and I would like to find the values of $e$ and $w$ as the solution of $\frac{\partial e}{\partial w}=\frac{e}{w}$. And my ultimate goal is to plot each of these values of $e$ and $w$ against $a$.
First, I found the $e$ function, $e=e(w)$ numerically for various values of $a$: $a = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1$ along with the other simulation values of $n=1$, $b=0.7$, $i=0.1$. My code for this is:
Clear["Global`*"]
n = 1;
eqns[i_, b_] = {D[w - 1/((1 - e) w), e] ==
D[1 - e, e] ((-((-a (i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(
1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) + (-1 +
a^2) (-1 + e)^2 w -
a (-1 + e) (i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(
1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) w^2)/(
a (-1 + e) i (1 - e + i + (
n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(
1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) w))) - (-((-a (
n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(
1 - n e (1/(b (a e)^(b)))^(1/(b - 1))) + (-1 + a^2) (-1 +
e) (-1 + e - i) w -
a (-1 + e) (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(
1 - n e (1/(b (a e)^(b)))^(1/(b - 1))) w^2)/(
a (-1 + e) i (1 - e + i + (
n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(
1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) w)))), 0 <= i <= 1, 0 <= a <= 1, 0 <= e <= 1, w >= 0, 0 <= b <= 1};
ContourPlot[Evaluate@ Table[Simplify@ eqns[1/10, 7/10][[1]], {a, {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1}}], {w, 0, 5}, {e, 0, 1}, PlotPoints -> 50, MaxRecursion -> 4, FrameLabel -> (Style[#, 14] & /@ {w, e}), RotateLabel -> False, PlotLegends -> Placed[LineLegend[{0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1},
LegendLabel -> "a"], {.7, .4}]]
Now, I would like to use this $e=e(w)$ function numerically found as above to solve $\frac{\partial e}{\partial w}=\frac{e}{w}$ and use the solutions for $e$ and $w$, which would be a function of $a$, to plot each of them against $a$. The plots I'm looking for would be something like these. Please help.