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.