f[x_] = Piecewise[{{-x, x <= 1}, {0, x > 0}}];
sol = DSolve[x'[t] == f[x[t]] + u, x[t], t]
(* {{x[t] -> InverseFunction[Piecewise[{{-Log[u - #1], #1 <= 1}}, -(1/u) - Log[-1 + u] +
#1/u] & ][t + C[1]]}}*)
Solution in implicit form:
((sol[[1, 1, 2, 0]][[1, 1]]) /. #1 -> x[t] // Simplify) == sol[[1, 1, 2, 1]]
(* Piecewise[{{-Log[u - x[t]], x[t] <= 1}}, (-1 - u*Log[-1 + u] + x[t])/u] == t + C[1] *)