# Solve coupled equations in the steady state

I want to solve 16 coupled differential equations in the steady state. Among the answers, I only require two of the answers. But the Mathematica does not show me the answers in detail!!!

My code is as follows: How can I get only the answer of v33 and v44 in details?

equil = {2 (-\[Kappa] + \[Lambda] Cos[\[Phi]]) v11 + (\[CapitalDelta] \
+ \[Lambda] Sin[\[Phi]]) (v12 + v21) +
g (v14 + v41) + \[Kappa]/
2 (1 + 2 \[ScriptCapitalN] +
2 \[ScriptCapitalM] Cos[\[Theta]]), -2 \[Kappa] v12 + (\
\[CapitalDelta] + \[Lambda] Sin[\[Phi]]) v22 + (-\[CapitalDelta] + \
\[Lambda] Sin[\[Phi]]) v11 +
g (v42 -
v13) + \[Kappa] \[ScriptCapitalM] Sin[\[Theta]], (-\[Kappa] - \
\[Gamma] + \[Lambda] Cos[\[Phi]]) v13 + (\[CapitalDelta] + \[Lambda] \
Sin[\[Phi]]) v23 +
g (v12 +
v43) + \[Delta] v14, (-\[Kappa] - \[Gamma] + \[Lambda] Cos[\
\[Phi]]) v14 + (\[CapitalDelta] + \[Lambda] Sin[\[Phi]]) v24 +
g (v44 -
v11) - \[Delta] v13, (-\[CapitalDelta] + \[Lambda] \
Sin[\[Phi]]) v11 + (\[CapitalDelta] + \[Lambda] Sin[\[Phi]]) v22 -
2 \[Kappa] v21 +
g (v24 -
v31) + \[Kappa] \[ScriptCapitalM] Sin[\[Theta]], (-\
\[CapitalDelta] + \[Lambda] Sin[\[Phi]]) (v12 + v21) -
2 (\[Kappa] + \[Lambda] Cos[\[Phi]]) v22 -
g (v23 + v32) + \[Kappa]/
2 (1 + 2 \[ScriptCapitalN] -
2 \[ScriptCapitalM] Cos[\[Theta]]), (-\[CapitalDelta] + \
\[Lambda] Sin[\[Phi]]) v13 + (-\[Kappa] - \[Gamma] - \[Lambda] Cos[\
\[Phi]]) v23 +
g (v22 -
v33) + \[Delta] v24, (-\[Delta] + \[Lambda] Sin[\[Phi]]) v14 + \
(-\[Kappa] - \[Gamma] - \[Lambda] Cos[\[Phi]]) v24 -
g (v34 + v21) - \[Delta] v23,
(-\[Kappa] - \[Gamma] + \[Lambda] Cos[\[Phi]]) v31 + (\
\[CapitalDelta] + \[Lambda] Sin[\[Phi]]) v32 +
g (v21 +
v34) + \[Delta] v41, (-\[CapitalDelta] + \[Lambda] \
Sin[\[Phi]]) v31 + (-\[Kappa] - \[Gamma] - \[Lambda] Cos[\[Phi]]) v32 \
+ g (v22 - v33) + \[Delta] v42, -2 \[Gamma] v33 +
g (v23 + v32) + \[Delta] (v34 + v43) + \[Gamma]/
2 (1 + 2 m), -2 \[Gamma] v34 +
g (v24 - v31) + \[Delta] (v44 -
v33), (-\[Kappa] - \[Gamma] + \[Lambda] Cos[\[Phi]]) v41 + (\
\[CapitalDelta] + \[Lambda] Sin[\[Phi]]) v42 +
g (v44 -
v11) - \[Delta] v31, (-\[CapitalDelta] + \[Lambda] \
Sin[\[Phi]]) v41 + (-\[Kappa] - \[Gamma] - \[Lambda] Cos[\[Phi]]) v42 \
- g (v12 + v43) - \[Delta] v32, -2 \[Gamma] v43 +
g (v42 - v13) + \[Delta] (v44 - v33), -2 \[Gamma] v44 -
g (v14 + v41) - \[Delta] (v34 + v43) + \[Gamma]/2 (1 + 2 m)};

Solve[equil == 0, {v11, v12, v13, v14, v21, v22, v23, v24, v31, v32,
v33, v34, v41, v42, v43, v44}]
• After a few minutes of waiting I could get a solution for all unknowns: {b, A} = CoefficientArrays[ equil == 0, {v11, v12, v13, v14, v21, v22, v23, v24, v31, v32, v33, v34, v41, v42, v43, v44}] followed by sol=LinearSolve[A, -b]followed by sol[[11]] and sol[[16]]. However, they are too long (with LeafCount being 730546 and 384903, respectively), and supposedly not very useful. – yarchik Apr 8 at 6:35
• Thank you so much for your useful response. – mehrosadat ebrahimi Apr 8 at 7:00