# To solve self consistent equations by simultaneously plotting them

How can we solve a pair of equations that have to be solved self-consistently (even by plotting them simultaneously) .

y = 1/2 + 1/\[Pi] ArcTan[c (0.5 - x)];
x = 1/2 + 1/\[Pi] ArcTan[c (0.5 - y)];


For c=0.5 we must have a plot as below one in which the vertical axis is y and horizontal one is devoted to x.

For c=Pi we must have

• FindRoot for solving and ContourPlot for plotting Commented Mar 27, 2019 at 10:06
• I wanted to plot the first function y vs x ordinary and by InverseFunction[f][y] I wanted to plot the second one (x vs y). For the second one the output plot is not symmetric relative to bisectrix of the first quarter of the coordination plate. Commented Mar 27, 2019 at 10:38

With[{c = 0.5},
ContourPlot[{
y == 1/2 + 1/π ArcTan[c (0.5 - x)],
x == 1/2 + 1/π ArcTan[c (0.5 - y)]
},
{x, 0, 1}, {y, 0, 1}]
]


With[{c = π},
ContourPlot[{
y == 1/2 + 1/π ArcTan[c (0.5 - x)],
x == 1/2 + 1/π ArcTan[c (0.5 - y)]
},
{x, 0, 1}, {y, 0, 1}]
]


• One can even use MeshFunctions to mark the intersection point. Commented Mar 27, 2019 at 13:36