# 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 – Lotus Mar 27 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. – Unbelievable Mar 27 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. – J. M. will be back soon Mar 27 at 13:36