Homework 2   Due February 5

Please note that there are lists of exercises on the page which gives the
lectures.  These are for the most part, problems as opposed to exercises.


1.  Solve the initial value problem  y'+y = exp(-rt) with y(0) = 0 for
r a constant greater than zero.  Pay special attention to r = 1. Show
that the solution is a continuous function of r as well as t.

2. page 61 #13

3. page 88 #3

4.  Page 91 #22 

5. Find the linearizations of the equation  y' = (y - 2) (1 - e^y) at the  
two fixed points, y = 0 and y = 2.



Extra Credit

Consider the differential equation  y' = y(ln(y) - ay)
for y non-negative.  Find the equilibrium points as a function of the
parameter a, identify the points which are stable and those which are
unstable, and the bifurcation points.  Sketch the bifurcation diagram.
indicate on the diagram which choices y(0) = y_0 generate trajectories
which approach the different stable equilibrium points. There is such
a diagram on page 93 of the text.