**Homework 8**

1) Boundary Value Problem:

(a) Solve the differential equation, **u_xx = - (pi^2/4)*(u +1 )** [Eq. 3.54
in the textbook], with the boundary conditions: **u_x(0) = 1** and **u_x(1)
= 1**. Use the shooting method. You may modify the program 'shooting.f' (or
its analogue in C). Make a plot of **u(x)** and **u_x(x)**. Obtain the
analytical result for this set of boundary conditions using Mathematica.

**(25 points) **

(b) Modify the program 'shooting.f' (or its analogue in C) to use equations
3.59 - 3.63 instead of the shooting method to solve the same problem given in
the book, **u_xx = - (pi^2/4)*(u +1 )**, with the boundary conditions: **u(0)
= 0** and **u(1) = 1**. Again, use 'shooting.f' (or its analogue in C) as
a starting point. Hand in a printout of your program, and mark the changes you
made to the original program.

**(25 points) **

2) Integral Equation:

Write a program to find **f(x)** in the interval from **x= -1** to **x=1**
from the integral equation:

**f(x) = x + 0.5 Integral[ dt (t - x) f(t) ]**,
where the integral limits are **-1** and **1**. You may use and modify
the program presented in class. Plot your solution **f(x)** in the interval
from **x= -1** to **x=1**, and give an analytical expression for it.
Please hand in a printout of your program.

**(50 points) **