# Solving a set of non-linear coupled differential equations

Here is a set of first-order differential coupled equations with

eqns = {x1'[
t] == (x4[t]^4 + (x4[t]^2 + x5[t]^2)^2 - x6[t]^4 +
x6[t]^2 x1[t] + x1[t]^2 + x3[t]^2 - x1[t] (x4[t]^2 + x5[t]^2)),
x2'[t] == (x2[t]^2 + x3[t]^2),
x3'[t] == x3[t]^2 (x1[t] + x3[t] - x4[t]^2 - x5[t]^2 + x6[t]^2),
x4'[t] == x4[t]^3,
x5'[t] == -x5[t]^3,
x6'[t] == x6[t] (x6[t]^2 - x4[t]^2 - x5[t]^2 - x7[t]^3),
x7'[t] == -x7[t]^3};
DSolve[eqns, {x1[t], x2[t], x3[t], x4[t], x5[t], x6[t], x7[t]}, t]


Mathematica returns an empty solution. Can anyone suggest a possible solution to it?

• You have a complicated, non-linear coupled system of 7 differential equations. There is very likely no analytic solution. Where did you get this system from? You can use NDSolve to get numeric solution. Commented Jun 10 at 12:58
• DSolve cannot even solve eqns[[{1,3}]] for {x1, x2} with the other dependent variables in those two equations set to 1, so it is very unlikely that it can solve the entire set. Note however that DSolve[eqns[[{4, 5, 6, 7}]], {x4, x5, x6, x7}, t] does return a solution. Commented Jun 10 at 17:46
• @Nasser: Excuse me, why have removed your answer? I have not even copied it. Could you please rewrite it. Txs Commented Jun 11 at 7:10