Homework 9
114B
due Wednesday November 27, 2002:
Do Boas Chapter 15:
3.4, 3.9, 6.1, 6.7, 8.6, 8.7, 8.18, 9.4, 9.5
1. Find the the general solution to G(x,x') in the equation
2. For 
, solve the equation
given that initially 
and that for all
one makes 
.
Hint: Laplace transform the equation with respect to time.
You now should get a differential equation involving only
an x derivative. Solve this equation and remember that it
can be multiplied by an arbitrary function of p (p is the
variable conjugate to t in the Laplace transform). To
determine this arbitrary function, Laplace transform the initial
condition 
. Now you'll need to find the inverse Laplace
transform. Remember that the Laplace transform of
is 
. Also the Laplace transform of 
is 
.
3. Consider the circuit for a low pass filter shown below.
is applied, and the output voltage is denoted V(t).
Show that V satisfies the equation
Recall that for a capacitor Q=CV, the voltage drop across a resistor is IR, and I = dQ/dt.
and
. Show that one can write
and solve for 
.
do obtain V(t) as
a convolution.
.
This is the Greens function for this problem.
. From this,
explain why this circuit is called a low pass filter.
4. The probability distribution function (PDF) is useful in many areas of science. The probability that a quantity has a value between x and x+dx is P(x) dx, where P(x) is the PDF of the quantity.
Events A and B are said to be independent
if the probability of finding A in the interval 
and
B in the interval 
is
Here the PDF for A is 
and the PDF for B is 
.
The PDF for 
is
where the S subscript denotes ``sum''.
as a convolution.
, where all the 
's are independent and
all have a PDF P(x), express the PDF of y as convolutions of P(x).
where 
(m is the mass and T is the temperature).
The velocity of different particles are independent of each other. Find the PDF of the sum
of the velocities of n particles 
in the x-direction. Hint:
use the convolution Theorem).