Differential voltage to current converter
The following simplifies all the calculations:







simulate this circuit &ndash; Schematic created using CircuitLab

Steps: 


Opamp terminals draw zero current. So the current directions are as above.
Write the equations for \$i_n\$ then solve for \$V_k\$ (1).
Write the equations for \$i_p\$ then solve for \$V_k\$ (2).
From (1) = (2), you'll find a nice relation between \$(V_p - V_n)\$ and \$(V_x - V_o)\$
Finally, since \$i_o=(V_x - V_o) / R_1\$, you'll obtain the relationship between \$i_o\$ and \$(V_p - V_n)\$.


I have the result, but don't ask me the full calculations. All the info you need are above.