HW #1, due Thursday January 29
==============================

Problem #1
----------

(a) By the method of separation of variables solve
the following IVP:


y'(t) = sqrt(y)

0 < t < 1, y is non-negative .

y(0) = alpha (a non-negative number).



(b) Find a solution corresponding to y(0)=0, using
the general solution from the part (a).
Also, guess a different solution to the problem with
y(0) = 0.

Now that you have two different solutions satisfying
the same initial condition, explain the seeming contradiction
to the Theorem 5.4 (p.252) that states that the solution
is unique.



Problem #2
----------

Write your own Euler solver, that will take
as an input a,b,n, alpha, and the name of the function
with f(t,y), and will solve approximately the ODE


y'(t) = f(t,y),

a<t<b,  y(a) = alpha,

using n intervals (or n+1 nodes).

(Some programming hints are given below.)

Solve the followng IVP using your solver:

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

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


Make several runs with n = 10, 20, 40, 80, 160.
Record the values of y at the last point (t=2).
(If recording manuallly, use the command "format long"
beforehand, to have enough digits printed).
Note that the exact solution to this equation is
y(t)=sqrt(t*t+2t+6) - 1.
Make a table that shows 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 2, since this is a first order method).

Please submit your code and the table.


The example below shows how to organize the connection between
the subroutines, so that you can solve IVP with different
right hand sides without changing the code of the solver:


File #1, call it myfunction.m:
--------------------------

function a = myfunction(b,c)

a = b*c;
end


File #2, call it callfunction.m
----------------------------

function callfunction(name,e,f)

d = feval(name,e,f);

display(d)
end



If both these files are present in the working directory,
the command

callfunction('myfunction',3,4);

will call the subroutine with the specified name ('myfunction')
and give it the parameters e and f (equal 3 and 4 in this case).
It will receive from the subroutine the output and will store
it in the variable d. In this simple example the output is just
a product of 3 and 4.