solving ordinary differential equations condition numericallywe more literally need an initial conditionwe need to know starting value at the left end of the interval in order to get the computational formula going in factwe do not need computer since we see repetitive pattern when doing hand calculationswhich leads us to mathematical formula for / / /td ktc rn ktd kt rd / / : /kc rather than using ( as computational model directlythere is strong tradition for deriving differential equation from this difference equation the idea is to consider very small time interval and look at the instantaneous growth as this time interval is shrunk to an infinitesimally small size in mathematical termsit means that we let as ( standsletting will just produce an equation so we have to divide by and then take the limitn td rn tt ! lim the term on the left-hand side is actually the definition of the derivative /so we have td rn /which is the corresponding differential equation there is nothing in our derivation that forces the parameter to be constant it can change with time due toe seasonal changes or more permanent environmental changes detourexact mathematical solution if you have taken course on mathematical solution methods for differential equationsyou may want to recap how an equation like rn or / is solved the method of separation of variables is the most convenient solution strategy in this casen rn dn rn dt dn rdt zn zt dn rdt