Examples

1. Plot the solutions of y'= y*(1-y). Why is y=1 a stable solution to the equation and y'=0 an unstable solution.
2. Plot the solutions to y'=(exp(x)-y)/x. Describe how the solutions appear to behave as x goes to infinity and as x goes to 0 from the right. Estimate the value for y(1) which gives a bounded solution as x goes to 0 from the right. Solve the equation analytically and determine this value exactly.
3. Consider the predator-prey equations, x' = a*x-b*x*y; y' = -c*y+d*x*y. They are meant to model two populations where population y depends on x as its food supply. Plot the solutions in the region x>0, y>0 for the parameter values a=3, b=2, c=5, d=3. What do you conclude about the long term behavior of the populations. Determine the equlibrium solution.
4. Consider the initial value problem y'=3*x^2, x'=3y^2-4, y(1)=0. Determine the largest interval on which this problem has a single-valued solution. Solve the equation analytically and determine the interval exactly.
5. Plot the solutions to Van der Pol's equation, x'=y, y'=-x+(1-x^2)*y.
6. Plot the solutions to x'=sin(y)-2*sin(x^2)*sin(2*y), y'=-cos(x)-2*x*cos(x^2)*cos(2*y) in the region -6 <= x <= 6, -6 <= y <= 6. Can you see why this is called the "teddy bear" equation?
7. Describe the behavior of solutions to the Ricati equation, y'= -y^2-x*y+1. Why do all solutions passing thru a point between the two y-nullcline curves converge to the nullclines after first crossing them?
8. Plot nullclines and solutions to y'= x*sin(x*y) in the window 0 <= x <= 10, 0 <= y <= 5. Explain what you see. (Be a little patient. There are a lot of nullclines and plotting them may take some time.) Now do the same thing in the window 0 <= x <= 20, 0 <= y <= 5. (In this case the nullclines are so close together the program misses most of them.)