Numerical Solutions of Differential Equations

Part 3: The Logistic Equation

1. Now we will obtain the solution to the initial value problem
   

dy/dt = y (1 - y), with y(0) = 0.1.

Plot the direction field, and then use your helper application's built-in differential equation solver to plot the solution to the initial value problem.

2. Find an exact symbolic solution of the initial value problem. (You may do this entirely with pencil and paper, or you may use your helper application for help with the algebra.) Check your symbolic solution by plotting it in your worksheet and comparing it with the approximate solution in step 1.
3. Change the initial condition to y(0) = 1.5, and repeat steps 1 and 2. Do your symbolic solutions correspond to the approximated solutions?