Consider: $$\frac{1}{(1-e)^2 w}=v-z$$ where $v=f(e,i,L,W,a,n)$, $z=g(e,i,L,W,a,w,n)$, and $L = h(e,w,a,b)$ are given functions.
From the above equation, $e$ can be derived as a function of $w$ along with the other parameters, i.e., $e=e(w;\dots)$. Using the latter, we can consider the following differential equation: $$\frac{\partial e}{\partial w}=\frac{e}{w}$$ I would like find $e$ and $w$ that satisfies the above differential equation. Obviously, both $e$ and $w$ will come out as a function of $a$ along with the other parameters.
Finally, I would like to create four separate plots for $e$, $w$, $\frac{e}{w}$, and $\frac{ae}{w}$ each of them against $a$ with the parameter values of $n=1$, $W=60$, $i=0.1$, and $b=0.6$.
By closely referring to Michael E2's answer to this post, I tried to come up with the following code:
Clear["Global`*"];
n = 1; W = 60;
L = (w/(b (a e)^b))^(1/(b - 1));
v = -(((-1 + e - i) (-i + (-1 + e + e i) L n) (1/((-1 + e) (-1 + e - i) w) + w/(-1 + e - i) + ((-1 + e) (1 - e L n) (-(1 + i)^2 (-1 + (1 + i)^((1 - e L n)/(L n - e L n)))^2 + a^2 i^2 (-1 + (1 + i)^(1 + 1/(L n - e L n)))^2 W^2))/(a i (1 + i) (1 - e + i) (1 - (1 + i)^((1 - e L n)/(L n - e L n))) (1 - (1 + i)^(1 + 1/(L n - e L n))) (i - (-1 + e + e i) L n) W)))/(i (1 + i + L n + e (-1 + (-2 + e - i) L n))));
z = -((a i (1 + i)^((1 + e)/(1 - e)) (-1 + (1 + i)^((1 - e L n)/(L n - e L n))) (-1 + (1 + i)^(1 + 1/(L n - e L n))) L n W + a (-1 + e) i (1 + i)^((1 + e)/(1 - e)) (-1 + (1 + i)^((1 - e L n)/(L n - e L n))) (-1 + (1 + i)^(1 + 1/(L n - e L n))) L n w^2 W + (-1 + e - i) (1 + i)^(-((2 e)/(-1 + e))) (-1 + e L n) w (1 + 2 i + i^2 - a^2 i^2 W^2 - 2 (1 + i)^(1 + 1/(L n - e L n)) ((1 + i)^(2 + 1/(-1 + e)) - a^2 i^2 W^2) + (1 + i)^(2 + 2/(L n - e L n)) ((1 + i)^((2 e)/(-1 + e)) -a^2 i^2 W^2)))/(a i^2 (1 + i) ((1 + i)^(e/(1 - e)) - (1 + i)^(1/(L n - e L n))) ((1 + i)^(e/(1 - e)) - (1 + i)^((1 + L n)/(L n - e L n))) (1 + i + L n + e (-1 + (-2 + e - i) L n)) w W));
myEQ = 1/((1 - e)^2 w) == v - z;
Block[{n = 1, W = 60, i = 1/10, b = 6/10, a = 1/2}, {#, Dt[#, w]} &@myEQ] /. {e -> e[w]} /. {e'[w] -> e[w]/w} /. {e[w] -> e} // Simplify;
icEQ = % /. Equal -> Subtract // Simplify;
icSOL = NSolve[icEQ == 0 && 0 < e < 1 && 0 < w < 2, {e, w}]
NDSolveValue[{ode = Solve[Block[{n = 1, W = 60, i = 1/10, b = 6/10}, D[{myEQ /. Equal -> Subtract, {e, w} . D[myEQ /. Equal -> Subtract, {{e, w}}]} /. {w -> w[a], e -> e[a]}, a] == 0], {e'[a], w'[a]}] /. Rule -> Equal, e[1/2] == (e /. First@icSOL), w[1/2] == (w /. First@icSOL)}, {e, w}, {a, $MachineEpsilon, 1}]
ListLinePlot[{e\[FivePointedStar], w\[FivePointedStar]}, PlotLegends -> Block[{e\[FivePointedStar], w\[FivePointedStar]}, HoldForm /@ {e\[FivePointedStar], w\[FivePointedStar]}], PlotRange -> All]
The code runs forever. Also, I failed to come up with a code that creates the four separate plots.
