deg = 5;
Coefficient[AsymptoticDSolveValue[ode, y[x], {x, 0, deg + 1}], x, deg] -
Coefficient[AsymptoticDSolveValue[ode, y[x], {x, 0, deg}], x, deg] // Simplify
Coefficient[AsymptoticDSolveValue[ode, y[x], {x, 0, deg + 2}], x, deg] -
Coefficient[AsymptoticDSolveValue[ode, y[x], {x, 0, deg + 1}], x, deg] // Simplify
(*
(C[1] (39 - 20 Log[x]))/9600
0
*)