In this note, we formulate a ``one-sided'' version of Wormald's differential equation method. In the standard ``two-sided'' method, one is given a family of random variables which evolve over time and which satisfy some conditions including a tight estimate of the expected change in each variable over one time step. These estimates for the expected one-step changes suggest that the variables ought to be close to the solution of a certain system of differential equations, and the standard method concludes that this is indeed the case. We give a result for the case where instead of a tight estimate for each variable's expected one-step change, we have only an upper bound. Our proof is very simple, and is flexible enough that if we instead assume tight estimates on the variables, then we recover the conclusion of the standard differential equation method.

翻译：本文提出了Wormald微分方程方法的“单侧”版本。在标准的“双侧”方法中，我们考虑一族随时间演化的随机变量，这些变量需满足若干条件，包括对每个变量单步期望变化的精确估计。这些单步期望变化的估计表明变量应接近某微分方程系统的解，而标准方法最终证明了这一结论。我们给出了一种仅需变量单步期望变化上界（而非精确估计）时的结果。我们的证明十分简洁，且具有足够的灵活性：若假设变量具备精确估计，则可还原标准微分方程方法的结论。