Example 3.  Lotka-Volterra Model.  Solve the I.V.P.  
      with  ,  
and
      with  .    
Combine the system of D. E.'s to form a separable first-order differential equation and solve the D. E..

Solution 3.

Write down the two equations.

Use the Leibniz notation for the derivative.

Divide the left sides and divide the right sides using the idea that .  

Separate the variables.

Integrate both sides and supply a constant of integration in the form  Log[c].

Simplify the form of the solution.

Solve for the constant  c  using the fact that the curve goes through the initial point  (2,1).

Replace the constant and form the implicit equation for the solution curve.

Graph the curve using Mathematica's the graphics package ImplicitPlot.

(c) John H. Mathews 2004