Assignment 9
Due November 7, 1996
Use Maple or Mathematic on the NeXT.
1. Solve the second-order ODE
y'' + 3y' + 2y = exp(-x)
(y'' and y' indicate derivatives wrt x)
Now solve the same ODE with the initial conditions
y(0) = 1 and y'(0) = 0
Plot the solution to the IVP over the range (0,8)
2. Consider the 3 by 3 Toeplitz matrix with elements a, b, and c.
(That is the matrix that looks like this:
a b c
b a b
c b a)
Invert this matrix.
Using Maple's solve() on certain expressions in the symbolic
solution you obtained, determine conditions for which the matrix
would be singular.
(Hint to reduce your typing: you can use such things as %1 or %2
to represent certain things.)
3. According to Maxwell-Boltzman theory, the probability density of
the velocity of a gas molecule is proportional to
(m/(k*T))**(3/2) * exp(-(m*v**2)/(2*k*T)) * v**2,
where v is the velocity, T is the absolute temperature, m is the
molecular mass, and k is Boltzman's constant.
Determine the mode of this distribution (the point where it
achieves its maximum value -- the "most likely" velocity) by
determining the critical points (points where the first
derivative is 0) and finding one of those whose second
derivative is ...(positive or negative?). Your solution is
called the rms velocity.
4. Produce a 3-d display of the "Sidney Opera House"; that is, plot the
function
sqrt{ Gamma(0.01+ |x|) * J_0(y) * J_1(y) },
over the ranges x:(-6,6) and y:(-3,3),
where $\Gamma$ is the gamma function and $J_i$ is the Bessel
function of the first kind, of order $i$. (Use ?Bessel and/or
?inifcns to get more info about these functions in Maple. Use
?plot3d also.)
Link this image into your Web page for this class.
Copyright John
Wallin 1996. All rights reserved.
Last Modified : Thur Aug 29 12:31:00 EST 1996
<jwallin@gmu.edu>