Eliminate a variable from two equations

Question

I have a set of two equations with three variables u,o,d. I want to eliminate either u or d from these equations.

2 (1 + u^2/d^2)^(1/4) == Sqrt[d] ((1 + (o - u)^2/d^2)^(1/4) + (1 + (o + u)^2/d^2)^(1/4))


2/3 (1/d)^(3/2) == (1/(4 d)) \[Pi] (o (-(1 + (o - u)^2/d^2)^(1/4) + (1 + (o + u)^2/d^2)^(1/4)) + u ((1 + (o - u)^2/d^2)^(1/4) + (1 + (o + u)^2/d^2)^(1/4)) + d ((1 + (o - u)^2/d^2)^(3/4) - (1 + (o + u)^2/d^2)^(3/4)))

So far I have tried using Eliminate, Reduce, Solve. After failing to get any output after several minutes, I eventually have to abort the evaluation. I cannot even seem to simplify these equations further. Any idea on how I can proceed? (Edit: My goal is to obtain a plot of u and d vs o.)

Answer 1

To obtain plots, you can use NDSolveValue to generate InterpolatingFunction solutions for d[o] and u[o]. First, here are your equations:

eq1 = 2 (1+u^2/d^2)^(1/4) == Sqrt[d] ((1+(o-u)^2/d^2)^(1/4)+(1+(o+u)^2/d^2)^(1/4));

eq2 = 2/3 (1/d)^(3/2)==(1/(4 d)) \[Pi] (o (-(1+(o-u)^2/d^2)^(1/4)+(1+(o+u)^2/d^2)^(1/4))+u ((1+(o-u)^2/d^2)^(1/4)+(1+(o+u)^2/d^2)^(1/4))+d ((1+(o-u)^2/d^2)^(3/4)-(1+(o+u)^2/d^2)^(3/4)));

To use NDSolveValue, we will need an initial condition. So

{di, ui} = {d, u} /. FindRoot[{eq1, eq2} /. o->-1, {{d, .5}, {u, .1}}]

{0.384684, 0.144757}

Next, we need to turn your equations into an ODE, and use NDSolveValue:

{dsol, usol} = NDSolveValue[
    {
    D[eq1 /. {d->d[o], u->u[o]}, o],
    D[eq2 /. {d->d[o], u->u[o]}, o],
    u[-1] == ui,
    d[-1] == di
    },
    {d, u},
    {o, -5, 2}
];

Finally, here's a visualization of the result:

Plot[{dsol[t], usol[t]}, {t, -5, 2}]