Solving for v_out in a non-ideal difference op amp
There are only three nodes where the voltages are not already determined. So, three equations are needed.

Non-inverting input:
$$
\frac{\text{Vp}-\text{V1}}{R}+\frac{\text{Vp}}{R}+\frac{\text{Vp}-\text{Vn}}{\text{Ri}}=0 
$$
Inverting input:
$$
\frac{\text{Vn}-\text{V2}}{R}+\frac{\text{Vn}-\text{Vout}}{R}+\frac{\text{Vn}-\text{Vp}}{\text{Ri}}=0
$$
And the output:
$$
\frac{\text{Vout}-A (\text{Vp}-\text{Vn})}{\text{Ro}}+\frac{\text{Vout}-\text{Vn}}{R}+\frac{\text{Vout}}{\text{RL}}=0
$$
Using a computer algebra program to solve this system yields
$$
\text{Vout} = \frac{\text{RL} (2 A R \text{Ri} (\text{V1}-\text{V2})+\text{Ro} (R (\text{V1}+\text{V2})+2 \text{Ri} \text{V2}))}{R (2 \text{Ri} ((A+2) \text{RL}+2 \text{Ro})+3 \text{RL} \text{Ro})+4 R^2 (\text{RL}+\text{Ro})+2 \text{Ri} \text{RL} \text{Ro}}
$$

This reduces to Vout = $V1-V2$ for \$Ro = 0\$ and \$Ri = \infty\$