The coagulation equation, which is widely used for modeling growth in planet formation and other astrophysical problems, is the mean-rate equation that describes the evolution of the mass spectrum of a collection of particles due to successive mergers. A numerical code that can yield accurate solutions to the coagulation equation with a reasonable number of mass bins is developed, and it is used to study the properties of solutions to the coagulation equation. We consider limiting cases of the merger rate coefficient A i j for gravitational interaction, with the power-law index of the massradius relation Ξ² = 1/3 (for planetesimals) and 2/3 (for stars). We classify the mass dependence of A i j using the exponent Ξ» for the merger between two particles of comparable mass, and the exponents Β΅ and Ξ½ for the merger between a heavy particle and a light particle. For the two cases with Ξ½ β€ 1 and Ξ» β€ 1, the mass spectrum evolves in an orderly fashion. For the remaining cases, which have Ξ½ > 1, we find strong numerical and analytical evidence that there are no self-consistent solutions to the coagulation equation at any time. The results for the Ξ½ > 1 cases are qualitatively different from the well-known example with A i j β ij. For the latter case, which is in the range Ξ½ β€ 1 and Ξ» > 1, there is an analytic solution to the coagulation equation that is valid for a finite amount of time t 0 . We discuss a simplified merger problem that illustrates the qualitative differences in the solutions to the coagulation equation for the three classes of A i j . Our results strongly suggest that there are two types of runaway growth. For A i j with Ξ½ β€ 1 and Ξ» > 1, runaway growth starts at t crit β t 0 , the time at which the coagulation equation solution becomes invalid. For A i j with Ξ½ > 1, which include all gravitational interaction cases expected to show runaway growth, the dependence of the time t crit for the onset of runaway growth on the parameters of the problem is not yet well understood, but there are indications that t crit (in units of 1/(n 0 A 11 )) may decrease slowly toward zero with increasing initial total number of particles n 0 .