U = DSolveValue[ {D[u[t] , t] - (F/m) + (b/m)*u[t] == 0  }, u, t]
(*Function[{t}, F/b + E^(-((b t)/m)) C[1]]*)

X = DSolveValue[{x[t] x'[t] == U[t]   }, x, t]
(*Function[{t}, -((Sqrt[2] Sqrt[F t - E^(-((b t)/m)) m C[1] + b C[2]])/Sqrt[b])]*)

{X[0], X'[0]}
(*{-((Sqrt[2] Sqrt[-m C[1] + b C[2]])/Sqrt[b]), -((F + b C[1])/(Sqrt[2] Sqrt[b] Sqrt[-m C[1] + b C[2]]))}*)