HW #4, due Thursday March 5
===========================


Consider IVP

y'(t) = lambda*(y(t)-cos t )
0 < t < 5
y(0) = A;

A = lambda^2 / ( 1+ lambda^2 ),
lambda = -80.


1. ==================================

By analyzing the test equation

y'(t) = lambda*y(t)

figure ot the largest time step hMax for
the Euler's method that would guarantee
the stability. What is the minimal number
Nmin of steps that would still give a
stable result?


Solve the above IVP by the Euler method
with the number of steps Nmin-5 and Nmin.
Plot these solutions versus the exact
solution given by the formula

y(t) = A*cos(t) - A*sin(t)/lambda.

Compute the maximum pointwise error in
these two runs.

2. ==================================

Compute approximate solution to the
above IVP using the (implicit) backward
Euler method. (You don't need to write
a general algorithm; just use the simplicity
of the equation to solve for w(k+1) on
each step). Compare your solution to the
exact solution. Play with different number
of steps  (different step sizes), and find
the smallest number of steps required
to achieve the (pointwise) accuracy of
0.005. (I mean the absolute error;
no need to compute relative values).

3. ==================================

What is the ratio of the number of steps
in forward and backward Euler methods
required to achieve the accuracy of 0.005?


4. ==================================

Solve problem 7, section 5.10.
(Hint: apply the method to the test equation y'(t) = 0 )


5. ==================================

Find the region of absolute stability for
the backward Euler's method. Is this method A-stable?