1.
(a)Following hint, try y = xp. Plugging into ODE:
(b) The general solution is then y = C+xp+ + C-xp-, and at x=1, y=0. Using this we obtain C+ + C- =0. Therefore .
2. Write
y(x,t) = X(x)T(t) and plugging into
(b) At x=2, y=0, so , which implies .
(c) Solving for , for .
(d) Summing over all solutions:
3.
and
Try
.
This implies
4. (a) From 1(b) for x < x'.
(b) Rescale x: , with the boundary condition y(x/2) = 0. Therefore C+ + C- =0, and so . for x> x'.
(c)
(e) .