# numerical solution of an equation containing an integral

I have the following problem: given r, I have to find the {x,y} which satisfies:

Integrate[1/(E^((2*x^2-4*x*d+2*d^2+2*y^2)/(8*d+4*r^2))*(Sqrt[d]*(2*d+r^2))),{d,0,Infinity}]==1


However, of all the solutions, I am interested to find the one which has the maximum y. Unfortunately, the integral can be solved only numerically. Any help? Bye

Many tahnks. It works now. However the calcualtion looks quite unstable. Have a look to my .nb:

this is the mathematica .nb file:

 rangeρ = {0, 0.1, 1, 10, 100};
rangeRy = {10^-5, 10^-4, 10^-3, 10^-2, 10^-1, 10^0, 10^1, 10^2, 10^3,
10^4, 10^5};
integrale[ξ_, ψ_, ζ_, ρ_] :=
NIntegrate[ (
E^(-((2 ξ^2 - 4 ξ δ + 2 δ^2 + 2 ψ^2 +
2 ζ^2 + (ρ^2 ζ^2)/δ)/(
8 δ + 4 ρ^2))))/(
Sqrt[δ] (2 δ + ρ^2)), {δ, 0, Infinity}];
solu = Table[
NMaximize[ψ,
2/Sqrt[π] Ry integrale[ξ, ψ, 0, ρ] ==
1, {ξ, ψ}][[2, 2, 2]], {ρ, rangeρ}, {Ry,
rangeRy}]


## for example, the large numebr obtained fro rho = 0, do not have much sense

I beg your pardon but I cannot manage to paste a notebbok as you did before. My code is:

ranger = {0, 0.1, 1, 10, 100};
rangeRy = {10^-5, 10^-4, 10^-3, 10^-2, 10^-1, 10^0, 10^1, 10^2, 10^3, 10^4, 10^5};

integrale[x_,y_,z_,r_]:=NIntegrate[(E^(-((2 x^2-4 x d+2 d^2+2 y^2+2 z^2+(r^2 z^2)/d)/(8 d+4 r^2))))/(Sqrt[d] (2 d+r^2)), {d,0,Infinity}];
solu=Table[NMaximize[y,2/Sqrt[π] Ry integrale[ξ,ψ,0,r]==1,{ξ,ψ}][[2,2,2]],{r,ranger},{Ry,rangeRy}]


If you willl run it you will see very big numbers that are surely a mistake. Problably there is the need of increasing the accuracy or the working precision? many thanks

• Please edit to add links to-from the cross-post on Wolfram Community. – Daniel Lichtblau Feb 6 '19 at 15:15

int[x_, y_, r_] :=NIntegrate[1/(E^((2*x^2 - 4*x*d + 2*d^2 + 2*y^2)/(8*d + 4*r^2))*(Sqrt[d]*(2*d + r^2))), {d, 0, Infinity}]

Now you can maximize as intended. For example r=1
NMaximize[{y , int[x, y, 1] == 1}, {x, y}]