(x[t] - x0) == a*(x'[t])^2 + b*x'[t] + c

$Assumptions =  a [Element] Reals && b [Element] Reals && c [Element] Reals

sol = DSolve[{-l0 + x[t]== a*(x'[t])^2 + b*x'[t] + c, x[0] == l0}, x[t], t]