Dear Vagif, real problems can have different nature but same logical structure, therefore they can be modeled by same system of equations. This reasoning is correct if the structure is the same, as in your examples; if structure has (even little) differences, the system of equations could be different. For more details, see Giorgio Israel, The Invisible Hand. Economic Equilibrium in

the History of Science, free available in RG archives.  Gianluca

For as long as two situations which are modelled have similar characteristic features, the same system of differential equations could be successfully used to analyze them. For instance, the spread of a number of infectious diseases is similar to the way rumours are spread - people need to have contacts in either case and one can also easily draw parallels between infected individuals and those who have already heard the rumor. Of course, every model is just a suitable, preferably the simplest approximation for the real situation which answers the questions one poses. Changing the questions one may need to change or modify the model too. Therefore, if more factors are taken into account, more complex models become and thus diverge more and more from the simplified model used for both situations in the very beginning. For instance, Lotka-Volterra model served its purpose but is not taking into account many factors such as, for instance, age, harvesting, or natural disasters. It can be further elaborated to incorporate new desirable features, but its analysis will be more and more elaborated to the point when analytic methods are no longer available and only numerical simulation can be used. Modeling process is endless, models can be indefinitely improved, and there exists no perfect model to perfectly match the complex situation which is modeled As George Box pointed out, "all models are wrong but some are useful."

The answer is obviously YES. However, somemtimes one can get impression that the mathematical models are different even if the differential equations are the same. This is probably caused by the already possessed specific knowledge about one phenomenon which is different if compared to the other processes. For instance, the functions to be sought are of different sign, and this implies different type of solutions, say in the first region they could be periodic and in the other one simply expanding to infinity (roughly speeking). Also, equations could have the same algebraic form, with different parameters. The impression of difference between the  methods can be strengthened due to difference in the order of difficulty in the different regions of initial conditionas and/or parameters! EVEN IF THEY SATISFY THE SAM EQUATION.

Thus, not only the differential equation counts, but also initial/boundary conditions. However, if the equations are exactly the same and the initial/boundary conditions are  exactly the same, then exactly the same bundle of  methods is applicable.

Of course, the choice of  particular equation and the method to a particular problem is an art for itself, and often requires deep understanding both - of the phenomenon and the of the math behind. Also the interpretation of particular variables of the assumed model can be better or  worse understandable by the researcher. This is already, in part at least, a domain of psychology of the mathematico-scientifcal abilities of the clever human kind.

Regards