HW #1, due Tuesday February 10 (I added a second problem !)
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Problem #1
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Write two more differential solvers.
One should use the second order modified
Euler method (aka trapezoid method),
the other should implement
the fourth order Runge-Kutta (RK4).


Solve the same problem as in HW 1, i.e.:

y'(t) = (1+t)/(1+y)

1<t<2,   y(1) = 2.

Make several runs with the number of steps
n = 10, 20, 40, 80, 160.
Record the values of y at the last point (t=2).
(If recording manually, use the command "format long"
beforehand, to have enough digits printed).
The exact solution to this equation is
y(t)=sqrt(t*t+2t+6) - 1.

Make two tables (one trapezoid, and one for RK4)
that show the error at the last point
against the number of steps, and the ratio showing
how the error improves each time the number of steps
is doubled. (We expect the improvement by a factor
of approximately 4 for trapezoid method, and by factor
of 16 for the RK4, since these are second- and forth-
order methods, respectively.)

Please submit your code (all subroutines) and the tables.


Problem #2
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Starting with the first formula on p. 293
(between (5.27) and (5.28) ) derive the (explicit)
four step Adams-Bashforth method. For simplicity,
you may drop the error term, and you can use table 5.10.
The answer is on page 296, so for a credit you need
to show your work.