1. (10 points)(a) Find the general solution to the equation
where 
and is a constant. Hint: Try a power law.
(b) What is the solution when 
? Use the
fact that 
and 
, to express your
result in terms of sine, logarithm, and 
.
2. (30 points) Consider the problem of a string with a line density that is
proportional to 
along its length. It obeys the equation
y(x,t) is the vertical displacement as a function of time. In this problem
1 < x < 2 and the the string is tied at the ends 
.
Find the frequencies of vibration as follows.
(a) Separate variables, using 
. The equation you
get for T should be 
.
(b) Solve the equations for T(t) and X(x), and apply the boundary condition y(x=1,t)=0. Hint: see problem 1.
(c) Apply the boundary condition y(x=2,t)=0, to determine what values
of 
are possible.
(d) Write down the general solution for y(x,t) as a sum over the solutions X(x)T(t).
(e) Draw a rough sketch of the lowest frequency mode, and also a sketch of a high frequency mode.
3. (10 points) An external force is applied to the string of problem 2 leading to the equation
F(x,t) is varies sinusoidally in time as
. By assuming
, find the differential
equation for X(x). is arbitrary and does not
have to be a frequency of vibration of the free string that
was obtained in 2(c).
4. (30 points) The equation obtained in problem 3 can be solved in general using Green's functions, by going through the following steps.
(a) First consider the case 
.
In this case 
.
Find the solution to the equation obtained in the last
problem for x < x' by applying the boundary condition
X(1) = 0 to the general solution of problem 1(a).
(b) Find the general solution for x > x' by applying the boundary condition X(2) = 0 to the general solution of problem 1(a).
G(x,x') is therefore of the form
where 
and 
were found in parts (a) and (b).
(c) By integrating the equation obtained in problem 3
over x from 
to 
, obtain
the the change in the derivative of G from x < x'
to x > x'.
(d) Using (b) and the fact that G is continuous, solve
for A(x') and B(x'). The answer is greatly simplified
by the identity 
.
(e) Now that the Green's function has been obtained, what is the general solution to equation obtained in problem 3, for any f(x)? (You don't have to write out the whole answer explicitly, just enough to make it clear that you know what you're doing).
5. (20 points) (a) How many different combinations of 5 cards out of a deck are there?
(b) What is the probability of obtaining a Royal Flush? (A Royal Flush is Ace, King, Queen, Jack, and 10, all of the same suite.)