Solve the differential equation
for the general initial condition at , where , and are positive constants. Deduce that the equilibria at and are stable and unstable, respectively.
By using the approximate finite-difference formula
for the derivative of at , where is a positive constant and , show that the differential equation when thus approximated becomes the difference equation
where and where . Find the two equilibria and, by linearizing the equation about them or otherwise, show that one is always unstable (given that ) and that the other is stable or unstable according as or . Show that this last instability is oscillatory with period . Why does this last instability have no counterpart for the differential equation? Show graphically how this instability can equilibrate to a periodic, finite-amplitude oscillation when .