Plot with implicit function

Question

This question is a follow-up from this post. Consider a function $e=f(w,a;i,\lambda)$ where $w$ is implicitly defined as $w=g(a;i,\lambda)$ such that $w\geq 0$. Hence $e$ is eventually a function of $a$, i.e. $e=f(a;i,\lambda)$.

The code for $e$ is

e == (w + a (-1 - a w + w^2) - \[Sqrt](a (a - (-1 + a^2) (i + lambda) w + a (i + lambda) w^2)))/(w + a w (-a + w))

And the code for the implicit function for $w$ is

-((a (-1 + a^2 - 2 a w) ((i + lambda) w - a^2 (i + lambda) w + a (2 + (i + lambda) w^2) + 2 \[Sqrt](a (a - (-1 + a^2) (i + lambda) w + a (i + lambda) w^2))))/(2 \[Sqrt](a (a - (-1 + a^2) (i + lambda) w + a (i + lambda) w^2)) (w + a w (-a + w))^2)) == 1/w ((w + a (-1 - a w + w^2) - \[Sqrt](a (a - (-1 + a^2) (i + lambda) w + a (i + lambda) w^2)))/(w + a w (-a + w)))

In the above link, I wanted to plot $e$ against $a$ with varying parameter values of $i \in [0,1]$ and $\lambda \in [0,1]$. This time, I would like to plot $\frac{e}{w}$ against $a$.

Applying the code from the answer (by Bob Hanlon) of the above link, my code for this question is:

eqns2 = {L == 1/w (w + a (-1 - a w + w^2) - \[Sqrt](a (a - (-1 + a^2) (i + lambda) w + a (i + lambda) w^2)))/(w + a w (-a + w)), -((a (-1 + a^2 - 2 a w) ((i + lambda) w - a^2 (i + lambda) w + a (2 + (i + lambda) w^2) + 2 \[Sqrt](a (a - (-1 + a^2) (i + lambda) w + a (i + lambda) w^2))))/(2 \[Sqrt](a (a - (-1 + a^2) (i + lambda) w + a (i + lambda) w^2)) (w + a w (-a + w))^2)) == 1/w ((w + a (-1 - a w + w^2) - \[Sqrt](a (a - (-1 + a^2) (i + lambda) w + a (i + lambda) w^2)))/(w + a w (-a + w))), 0 <= i <= 1, 0 <= lambda <= 1, 0 <= a <= 1, L >= 0, w >= 0};

sol2[i_, lambda_, a_] = SolveValues[eqns2, L, {w}, Reals] // Normal

Manipulate[Plot[Evaluate@sol2[i, lambda, a], {a, 0, 1}, Frame -> True, PlotRange -> {0, 1}, FrameLabel -> (Style[#, 14] & /@ {"a", "e/w"})], {{i, 0.1}, 0, 1, 0.01, Appearance -> "Labeled"}, {{lambda, 0.1}, 0, 1, 0.01, Appearance -> "Labeled"}]

It runs forever... I don't know why because the structure is exactly the same as in the above link.

Answer 1

To plot e/w versus a

Clear["Global`*"]

eqns[i_, lambda_] = {e == (w + 
       a (-1 - a w + 
          w^2) - √(a (a - (-1 + a^2) (i + lambda) w + 
            a (i + lambda) w^2)))/(w + 
       a w (-a + 
          w)), -((a (-1 + a^2 - 2 a w) ((i + lambda) w - a^2 (i + lambda) w + 
           a (2 + (i + lambda) w^2) + 
           2 √(a (a - (-1 + a^2) (i + lambda) w + 
                 a (i + lambda) w^2))))/(2 √(a (a - (-1 + a^2) (i + 
                 lambda) w + a (i + lambda) w^2)) (w + a w (-a + w))^2)) == 
    1/w ((w + 
         a (-1 - a w + 
            w^2) - √(a (a - (-1 + a^2) (i + lambda) w + 
              a (i + lambda) w^2)))/(w + a w (-a + w))), r == e/w, 
   0 <= i <= 1, 0 <= lambda <= 1, 0 <= a <= 1, 0 <= e <= 1, w >= 0};

Plotting,

Manipulate[
 Module[{i = Rationalize[iv], lambda = Rationalize[lambdav]},
   sol = SolveValues[eqns[i, lambda], r, {e, w}, Reals];
   Plot[Evaluate@sol, {a, 0, 1},
    PlotRange -> {0, 1},
    PlotRangePadding -> Scaled[.05],
    Frame -> True,
    FrameLabel -> (Style[#, 14] & /@ {"a", "e/w"})]] // Quiet,
 {{iv, 0.1, "i"}, 0, 1, 0.01, Appearance -> "Labeled"},
 {{lambdav, 0.1, "lambda"}, 0, 1, 0.01, Appearance -> "Labeled"},
 SynchronousUpdating -> False,
 TrackedSymbols :> All]