1. Consider the problem

(a) Explain how you know this problem is in the second alternative.

ans: y(x) = c is a non-trivial solution

to the non-homogeneous problem.

(b) Find linearly independent solutions for the equation y=**K***(y).

(c) Let f^{1}(x) = 3x - 1 and f^{2}(x) = 3x^{2} - 1. For one of these there
is a solution to the equation y = K(y) + f, for the other there is not. Which
has a solution?

ans: 3x^{2}-1 .

(d) For the f for which there is a solution, find two.

ans: 3x^{2 }-1 + 7 and 3x^{2}- 1 + 11.

2. Consider the problem

(a) Show that the associated **K** is small in both senses of this
section.

(b) Compute \phi_{2} where f(x) = 1. ans: 2 x^{2}/5 + 1

(c) Give an estimate for how much \phi_{2} differs from the solution y of
y=**K**(y)+f.

ans: error <= 1/(24 25^{2})

(d) Using the kernel k for **K**, compute the kernel k_{2} for
**K**^{2} and k_{2} for **K**^{3}.

ans: k_{2}(x,t) = x^{2}t^{2}/5.

(e) Compute the kernel for the resolvent of this problem.

ans: r(x,t) = 5x^{2}t^{2}/4

(f) What is the solution for y=**K**y+f in case f(x) = 1.

ans: y(x) = 1 + 5 x^{2}/12

3. Consider the problem

(a) Compute the associated approximations \phi_{0}, \phi_{1}, \phi_{2}, and
\phi_{3}.

ans: \phi_{1}(x) = x^{2} + x/6

(b) Give an estimate for how much \phi_{3} differs from the solution.

(c) Give the kernel for the resolvent of this problem.

ans: r(x,t) = 5xt^{3}/4

(d) Using the resolvent, give the solution to this problem.

ans: y(x) = x^{2} + 5x/24

(e) Using the fact that the kernel of the problem separates, solve the equation.

4. Suppose that

(a) Show that

(b) Solve the problem y = **K**[y] + 1. ans: y(x) = cos(x)/cos(1).

5. (a). Find a nontrivial solution for y = **K**[y] in L^{2}[0,1]
where

K(x,t) = 1 + cos(\pi x) cos(\pi t).

(b) Find a nontrivial solution for z = **K***[z].

(c) What condition must hold on f in order that

y = **K**[y] + f

shall have a solution. Does f(x) = 3 x^{2 }meet this condition.

ans: Constant functions are nontrivial solutions for both equations and the equation of IV(c) has a solution provided

The function 3 x^{2} does not meet this condition.

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