The first two questions are worth 2 points each, the last one 6 points
Find the Fourier series of a given function and then plug a
particular x to find the value of a certain infinite series
(compare Problems 60 and 62).
Prove a little statement concerning the solution of a certain
initial value problem (this is real similar to Problems 33 and 46).
Given is the diffusion equation on the interval from zero to one
together with certain boundary conditions. You have to perform
separation of variables and find all eigenvalues of the resulting
eigenvalue problem (it will be possible to give the eigenvalues
explicitely, as e.g., in Problem 55).
Then you have to decide whether it's boundary
conditions are symmetric. Finally, you have to find solutions
of this problem that satisfy at the same time a certain initial
condition u(x,0)=f(x). Well, this you have to do for three
different functions f. The first two are easy while the last one
is hard. Related problems from the homework assignments consist of
Problems 52--59.