How do I find the Millman theorem in this op-amp circuit?
Well, we are trying to analyze the following circuit (assuming an ideal opamp):



simulate this circuit &ndash; Schematic created using CircuitLab
When we use and apply KCL, we can write the following set of equations:
$$
\begin{cases}
\text{I}_1=\text{I}_2\\
\\
\text{I}_4=\text{I}_2+\text{I}_3\\
\\
\text{I}_4=\text{I}_1+\text{I}_3
\end{cases}\tag1
$$
When we use and apply Ohm's law, we can write the following set of equations:
$$
\begin{cases}
\text{I}_1=\frac{\text{V}_\text{b}-\text{V}_1}{\text{R}_1}\\
\\
\text{I}_2=\frac{\text{V}_1-\text{V}_3}{\text{R}_2}\\
\\
\text{I}_3=\frac{\text{V}_2-\text{V}_3}{\text{R}_3}\\
\\
\text{I}_4=\frac{\text{V}_3}{\text{R}_4}
\end{cases}\tag2
$$
Substitute \$(2)\$ into \$(1)\$, in order to get:
$$
\begin{cases}
\frac{\text{V}_\text{b}-\text{V}_1}{\text{R}_1}=\frac{\text{V}_1-\text{V}_3}{\text{R}_2}\\
\\
\frac{\text{V}_3}{\text{R}_4}=\frac{\text{V}_1-\text{V}_3}{\text{R}_2}+\frac{\text{V}_2-\text{V}_3}{\text{R}_3}\\
\\
\frac{\text{V}_3}{\text{R}_4}=\frac{\text{V}_\text{b}-\text{V}_1}{\text{R}_1}+\frac{\text{V}_2-\text{V}_3}{\text{R}_3}
\end{cases}\tag3
$$
Now, using an ideal opamp, we know that:
$$\text{V}_+=\text{V}_-=\text{V}_\text{a}=\text{V}_1$$
So we can rewrite equation \$(3)\$ as follows:
$$
\begin{cases}
\frac{\text{V}_\text{b}-\text{V}_\text{a}}{\text{R}_1}=\frac{\text{V}_\text{a}-\text{V}_3}{\text{R}_2}\\
\\
\frac{\text{V}_3}{\text{R}_4}=\frac{\text{V}_\text{a}-\text{V}_3}{\text{R}_2}+\frac{\text{V}_2-\text{V}_3}{\text{R}_3}\\
\\
\frac{\text{V}_3}{\text{R}_4}=\frac{\text{V}_\text{b}-\text{V}_\text{a}}{\text{R}_1}+\frac{\text{V}_2-\text{V}_3}{\text{R}_3}
\end{cases}\tag4
$$
Now, we can solve for \$\text{V}_2\$:
$$\text{V}_2=\text{V}_\text{a}\cdot\left(1+\frac{\text{R}_3}{\text{R}_4}\right)+\left(\text{V}_\text{a}-\text{V}_\text{b}\right)\cdot\frac{\text{R}_4\left(\text{R}_2+\text{R}_3\right)+\text{R}_2\text{R}_3}{\text{R}_1}\tag5$$
Where I used the following Mathematica-code:
In[1]:=V1 = Va; FullSimplify[
 Solve[{I1 == I2, I4 == I2 + I3, I4 == I1 + I3, I1 == (Vb - V1)/R1, 
   I2 == (V1 - V3)/R2, I3 == (V2 - V3)/R3, I4 == V3/R4}, {V2, V3, I1, 
   I2, I3, I4}]]

Out[1]={{V2 -&gt; ((R3 + R4) Va)/R4 + ((R2 + R3 + (R2 R3)/R4) (Va - Vb))/R1, 
  V3 -&gt; ((R1 + R2) Va - R2 Vb)/R1, I1 -&gt; (-Va + Vb)/R1, 
  I2 -&gt; (-Va + Vb)/R1, I3 -&gt; (R1 Va + (R2 + R4) (Va - Vb))/(R1 R4), 
  I4 -&gt; ((R1 + R2) Va - R2 Vb)/(R1 R4)}}