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 f¹(x) = 3x - 1 and f²(x) = 3x² - 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: 3x²-1 .

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

ans: 3x² -1 + 7 and 3x²- 1 + 11.

2. Consider the problem

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

(b) Compute \phi₂ where f(x) = 1. ans: 2 x²/5 + 1

(c) Give an estimate for how much \phi₂ differs from the solution y of y=K(y)+f.

ans: error <= 1/(24 25²)

(d) Using the kernel k for K, compute the kernel k₂ for K² and k₂ for K³.

ans: k₂(x,t) = x²t²/5.

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

ans: r(x,t) = 5x²t²/4

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

ans: y(x) = 1 + 5 x²/12

3. Consider the problem

(a) Compute the associated approximations \phi₀, \phi₁, \phi₂, and \phi₃.

ans: \phi₁(x) = x² + x/6

(b) Give an estimate for how much \phi₃ differs from the solution.

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

ans: r(x,t) = 5xt³/4

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

ans: y(x) = x² + 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 L²[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 x² 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 x² does not meet this condition.