# Plotting a=f(b) and b=g(a) on the (a,b) plane when neither has a closed solution

I've searched for a solution but haven't found one yet, but this is possibly an easy fix - I'm new to Mathematica. I have two functions, each with $$(a,b)$$ as arguments and both with no closed form expression. One is a condition on $$a$$, "given $$b$$", and the other is a condition on $$b$$, "given $$a$$". $$(1)\quad \frac{ S a^{\delta -1} b^{\delta }}{\left(a^{\delta }+b^{\delta }\right)^2}=1, \quad (2) \quad \frac{Sa^{\delta } b^{\delta -1}}{\left(a^{\delta }+b^{\delta }\right)^2}=1$$ I'm trying to plot both on a single plane, by using manipulate for S. Thanks to this question I am able to plot one of the equations (eq. $$(1)$$ which maps $$b \rightarrow a$$), by using Solve. This is my code.

\[Delta] = 1
line1 = Line[{{0, 0}, {2, 2}}]
Manipulate[ sol = a /.
Solve[(
a^(-1 + \[Delta])
b^\[Delta] S \[Delta])/(a^\[Delta] + b^\[Delta])^2 == 1, a];
Plot[sol, {b, 0, 2}, PlotRange -> {{0, 2}, {0, 1}},
Epilog -> {line1}], {S, 0, 2} , {\[Delta], 0, 2}]


The result is exactly what I would expect based on my own algebra and sketching skills. Now I need to add the second equation, which maps $$a \rightarrow b$$. I tried following the instructions in the question above, making sol={a,b} and solving two equations, one for $$a$$ and one for $$b$$, but it did not work. I'm guessing that has to do with $$b$$ being the "x-axis" for the Plot function...

Manipulate[ sol = {a, b} /.
Solve[{(
a^(-1 + \[Delta])
b^\[Delta] S \[Delta])/(a^\[Delta] + b^\[Delta])^2 == 1, (
b^(-1 + \[Delta])
a^\[Delta] S \[Delta])/(a^\[Delta] + b^\[Delta])^2}, {a, b}];
Plot[sol, {b, 0, 2}, PlotRange -> {{0, 2}, {0, 1}},
AxesLabel -> {b, a}, Epilog -> {line1}], {S, 0, 2} ]


Seeing as the equations are symmetric I thought maybe I could graphically transpose them. But I couldn't get it to work (with both lines on the same plot.) I'd also like to learn a general solution, for the future..

Hopefully this is clear, and thanks for any help.

• You can just use ContourPlot to plot implicit equations directly. – wxffles Nov 23 '18 at 1:43

Manipulate[