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I have two integral equations

-Sin[v]*Integrate[
     (x[v, z]*b*Sin[v])/Sqrt[(a^2*(Cos[v])^2 + b^2*(Sin[v])^2 + z^2)^3],
     {v, 0, 2*Pi}, {z, -l, l}] == 0

-Cos[v]*Integrate[
    (x[v, z]*a*Cos[v])/Sqrt[(a^2*(Cos[v])^2 + b^2*(Sin[v])^2 + z^2)^3],
    {v, 0, 2*Pi}, {z, -l, l}] == 0

I would like to find a solution for x[v,z], that is going to solve bough equations. Than I would like to plot 3D graph with axes v,z,x[v,z]. Can anyone help me in finding a solution?

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  • $\begingroup$ Cos and Sin are case sensitive cos[x] and sin[x] wont do anything. I updated the code in your post $\endgroup$ – ssch Sep 26 '13 at 11:01
  • $\begingroup$ I'm confused by the region of integration {z,-z,z}, could you double check this? $\endgroup$ – ssch Sep 26 '13 at 11:10
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There are infinitely many solutions satisfying the system of the equations.
If we set x[v, z] to be constant and add appropriate assumptions the both integrals vanish.
Moreover we could find a few consecutive values of the integrals for e.g. x[v, z] == z^k
and k ∈ {0, 1, 2, ...} putting it in a Table.
Nevertheless we can do much more proving an appropriate theorem:

If x[v, z] doesn't depend on v and it is a polynomial function in z then the both integrals vanish.

To get this result let's check

Resolve[ 
  ForAll[ k, k ∈ Integers && k >= 0, 
          Integrate[ z^k Cos[v]/Sqrt[(a^2 Cos[v]^2 + b^2 Sin[v]^2 + z^2)^3], 
                     {v, 0, 2 Pi}, {z, -l, l}, 
                     Assumptions -> l > 0 && (a | b) ∈ Reals] == 0]]
True

similarly

Resolve[ 
  ForAll[ k, k ∈ Integers && k >= 0, 
          Integrate[ z^k Sin[v]/Sqrt[(a^2 Cos[v]^2 + b^2 Sin[v]^2 + z^2)^3], 
                     {v, 0, 2 Pi}, {z, -l, l}, 
                     Assumptions -> l > 0 && (a | b) ∈ Reals] == 0]]
True

Since integration operator is linear we have proved that for any polynomial p[z] == x[v, z] the both integrals vanish. QED.

Calulation of the integrals if x[v, z] depends on v only is much more involved and we could find its values on a case by case basis. If x[v, z] == v z the system is satisfied under assumptions l > 0 && (a | b) ∈ Reals.

Let's plot a few special cases of given functions

With[{a = 1, b = 2}, 
  GraphicsRow @ Table[
    Plot3D[ v z^k Cos[v]/Sqrt[(a^2 Cos[v]^2 + b^2 Sin[v]^2 + z^2)^3], 
            {v, 0, 2 Pi}, {z, -2, 2}, ClippingStyle -> None], {k, 0, 2}]]

enter image description here

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