427K   First Computer Assignment   Due February 15    Uhlenbeck


        First Order Equations
             


 This is the first computer homework assignment. The exercise will
 be in two stages.  The assignment will be given first.  If you
 are already an expert at computation, you can do the problems any way
 you wish.

 For those who are not expert with either computers, or software for
 computation, the next page will walk you through using netmath on
 the mathematics department computers.  Later assignments will assume
 you have mastered this one.

PartI:

 IA:  Find the general solutions to the following equations using
 a computer program that uses symbolic manipulation.

  1. y' +  3*y  =   sin(4*x).

  2. y' - 3*y =  sin(4*x)).

  3. y' + 10*y =  sin(4*x).

  4. y' + .1*y =  sin(4*x).

  5. y' = .5*y*(3 - y).

 IB:  Now find the solutions to these equations with y = 1 at x = 0.

 IC:  Can you say anything in general about the equations?  What happens 
 to y as x goes to infinity.  What do you generally expect the solutions
  curves to do?

PartII:

 IIA:   Using a program which draws the direction  fields, draw the direction
  fields for these same five equations. Each time you draw the graph, go
  on to part B and plot the initial curve. 
 
IIB: Find the specific solution which is asked for in part B.  Be sure to
  adjust the parameters so that the important features of the curve are
  shown.  If necessary, graph only a short interval, but be sure to get
  the step size small enough to give a good curve. Hand in these graphs
  (this is the only printing you should need to do).

IIC: Can you improve the answer you got in IC now that you have drawn
  some graphs?