# Solving a system of integral equations

Is it possible to solve a system of integral equations in Mathematica? More specifically, I would like to obtain numerical solutions for $\mu$ and $\sigma$ from the following system:

f[x_] := (Sqrt[2*Pi]*σ*x*(1 - x))^(-1)*Exp[-0.5*σ^(-2)*(Log[x/(1 - x)] - μ)^2]
g[μ_, σ_] := Integrate[x*f[x], {x, 0, 1}]
h[μ_, σ_] := Integrate[x^2*f[x],{x, 0, 1}]
Solve[{g[μ, σ] == 0.3, h[μ, σ] == 0.1}, {μ, σ}]


The last line of code is a naive attempt to solve the system that does not run.

Help would be greatly appreciated!

• Note that Mathematica uses Log[] for the natural logarithm. – Feyre Jul 5 '16 at 9:22
• Oops! Have corrected it now. – Miguel Jul 5 '16 at 9:29

f[x_, μ_, σ_] :=

• @Miguel No, it is caused by symbolic expansion. If you feel this is annoying, first execute Clear[f,g,h] and then run my updated code, there should not be errors anymore. PS: that ?NumericQ prevents the equation from symbolic expansion, so NIntegrate won't complain. – vapor Jul 5 '16 at 9:51
• Thanks - that does get rid of the initial warnings. And changing the starting point for $\mu$ to 0 in your code gets rid of the remaining convergence warnings. Thanks again! – Miguel Jul 5 '16 at 10:03