Date: Fri, 3 Nov 95 14:37:18 CST From: rusin (Dave Rusin) To: tomwh@hiof.no Subject: Re: Integrating 2 simultanously dependent equations Newsgroups: sci.math In article you write: >dN1(t)/dt=(f1-m1*N1(t)-n1*N2(t))*N1(t) >dN2(t)/dt=(f2-m2*N2(t)-n2*N1(t))*N2(t) > >Is it possible to integrate these two equations and what the result be? If you want a closed form for N1 and N2 in terms of t, the answer is no -- not for general f1, f2, m1, n1, m2, n2 anyway. However, some things are possible sometimes. If you ignore t and look only at the relationship between the values of N1 and N2, you will see you get only certain combinations of N1 and N2 as time goes on. The curve joining all these posibilities is a solution to the equation dN1 (f1-m1*N1-n1*N2)*N1 --- =------------------- dN2 (f2-m2*N2-n2*N1)*N2 in the (N1, N2)-plane. Even this equation is in general not solvable in closed form, but certain cases are possible. For example, if m1=n2=0 or if n1=m2=0 then the equation is separable, so that there exists a solution of the form f(N1)=g(N2), which describes (implicitly) a curve in the (N1, N2)-plane. Moreover, even if no explicit equation describing the curve exists, a sketch of the curve may be given by marking at each point in the curve the slope the curve must have if if passes through that point. By joining together the marks, one obtains a sketch of the curve. This is all pretty standard: look up autonomous differential equations in a differential equations or modelling book. (This situation arises if, for example, N1 and N2 represent the populations of two species in a closed system.) dave