You are given an ordinary differential equation

3/'(t) = f(t,y), tela, b), where

f(t, y) and fy(t, y) are continuous and bounded in the vertical strip D = [a, b] x IR.

Given a number I, you are to find a E IR so that the solution to the IVP,

y'(t) = f(t, y), te [a, b], y(a) = a, satisfies

L." u(t) at = 1.

Just to be clear: you are given a < b E R, a subroutine that will evaluate f(t, y) for

(t, y) ED, and I E R. From this, you are to explain how to find a E R. You may

assume such a exists and is unique.

You can invent a new method or use methods we have discussed in class. Your

algorithm should be clear, but does not need the detail of source code. You may

assume the reader knows all of the methods we have discussed in class, so don't

describe any of them in detail.

You will be graded mostly on your discussion of why you selected the method(s) that

you did. Discuss possible sources of errors and when we might expect them to be

large. Analyze the algorithm to estimate which steps will use the most computational

resources (function evaluations and/ or memory). There are many different answers to

this problem that would get full credit.

Hint: Since y depends on a, one could write y = y(t; a), and you may want to

consider a function like g(a) = 1 - fa y(t; a) dt.

2. Theorem 5.10 (inequality 5.13) of our text gives an upper bound on the error from

Euler's method to (IVP) using the step size h and a bound, , on the rounding errors.

Assume you know a = 0, M = 4 and L = 3. Take o ~ 10-13 and do ~ 2 * 10-16 as

your o values. Using all of these values, find a value of h which minimizes the upper

bound on the absolute error given in the theorem at t = b. Now using this value of h,

and for each of b = 1, b = 2, 6 =4, and b = 8, give the upper bound for the error at

t =b, the number of time steps N, and the number of function evaluations needed.

Now in an attempt to model an RK4 method, replace the factor (by + ; ) (from the

theorem referenced above) with (Sh,M + " ), find a new optimal h, and redo your error

bounds, time steps, and function evaluations as above. Discuss your results briefly.