Together[Eqs - Simplify[Eqs]]

{{0.,0,0,0},{0,0.,0,0},{0,0,-((8. (1. (q4^[Prime])[x]^2-5. x^2 (q4^[Prime])[x]^2+10.25 x^4 (q4^[Prime])[x]^2-11. x^6 (q4^[Prime])[x]^2+6.25 x^8 (q4^[Prime])[x]^2+1. x^9 (q4^[Prime])[x]^2-3 vv.5 x^10 (q4^[Prime])[x]^2+1. x^11 (q4^[Prime])[x]^2))/((-2.+x^2)^2 (-1.+x^2)^2)),0},{0,0,0,0.}}

{[Tau], [Rho], x, [Phi]}

$Assumptions =And[[Tau] [Element] Reals, [Phi] > 0, [Phi] < 2*[Pi],[Rho] >
0, x > 0, x < 1, L > 0] ;