The Bessel equation of order p is
t2y" + ty' + (t2 - p2)y = 0. In
this problem, assume that p = 1/2:
a.) Show that y1 = sin(t / sqrt(t)) and y2
= cos(t / sqrt(t)) are linearly independent solutions for 0 < t
< infinity.
b.) Use the result from part (a), and the preamble in Exercise
3, to find the general solution of t2y" + ty' +
(t2 - (1/4))y = t3/2cos(t). (answer should
be: 1/2...
Consider the equation xy′′+y′+y= 0, x >0.
a) Verify that 0 is a regular singular point.
(b) Find the indicial equation and its roots.
c) Determine the recurrence relation(you do NOT need to find the
solutions).
a. Verify that the given point lies on the curve.
b. Determine an equation of the line tangent to the curve at the
given point.
9 (x2 y2)2 =100xy2 ;
(1,3)
For each pair a, b with a ∈ R − {0} and b ∈ R, define a function
fa,b : R → R by fa,b(x) = ax + b for each x ∈ R.
(a) Prove that for each a ∈ R − {0} and each b ∈ R, the function
fa,b is a bijection.
(b) Let F = {fa,b | a ∈ R − {0}, b ∈ R}. Prove that the set F
with the operation of composition of...
Suppose that the following processes arrive for execution at
time 0 in the order A, B, C:
process
Run Time
Priority
A
4
1=high
B
2
3=low
C
3
2=medium
Determine the execution order (with the time marks) of these
process using the following 4 schedulings: FCFS, SJF, a
non-preemptive priority, and RR (time slice = 2).
What is the waiting time of each process for each of the
scheduling algorithm? Which scheduling algorithm achieves the
shortest average waiting time?
Suppose that X ~ NB(r, p) [negative binomial distribution] and
that Y ~ B(n, p) [binomial].
a. Give a probabilistic argument to show that P(X > n) = P(Y
< r).
b. Use the FPF to express the equality in part (a) in terms of
PMFs.
c. Using the complementation rule, how many terms of the PMF of
X must be evaluated to determine P(X > n)?
d. How many terms of the PMF of Y must be evaluated to...
IS curve: Yt =a―b(Rt―r)
a=0, b=1, r=4%
Suppose that the central bank sets the real interest rate to 5%,
will this economie's level of short run output be above, below or
at potential output?
Work out the products of the matrices E, R, R^2,R^3, A, B, C,
D,
and verify that these products reproduce the multiplication table
for the
symmetries of the square, as expected. (Instead of computing all 64
products,
compute “sufficiently many” products, and show that your
computations
suffice to determine all other products.)
Consider the nonlinear equation f(x) = x3−
2x2 − x + 2 = 0.
(a) Verify that x = 1 is a solution.
(b) Convert f(x) = 0 to a fixed point equation g(x) = x where
this is not the fixed point iteration implied by Newton’s method,
and verify that x = 1 is a fixed point of g(x) = x.
(c) Convert f(x) = 0 to the fixed point iteration implied by
Newton’s method and again verify that...