Verify that it is the solution to the differential equation x^2 y'' + x y' + _x^2 y = 0, satisfying y(0)=1, y'(0)=0. Here y' means the first derivative of y(x) and y'' means the second derivative.

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Problem:

Verify that it is the solution to the differential equation x^2 y'' + x y' + _x^2 y = 0, satisfying y(0)=1, y'(0)=0. Here y' means the first derivative of y(x) and y'' means the second derivative.

Solution:

The general form of a Bessel equation is:
( 1)
and its general solution is
( 2)
where Bessel's function of first species of order "".
(Actually this is ...

Solution Summary

This solution is comprised of a detailed explanation to verify that it is the solution to the differential equation x^2 y'' + x y' + _x^2 y = 0, satisfying y(0)=1, y'(0)=0. Here y' means the first derivative of y(x) and y'' means the second derivative.

7-4
Consider the displacement of ,u(r,,t) , a "pie-shaped" membrane of radius a and angle /3 that satisfies:
utt = c22u
Assume that >0. Determine the natural frequencies of oscillation if the boundary conditions are:
Problem a.
a) u(r, 0, t) = 0, u(r, /3, t) = 0, ur(a, , t) = 0
proble

(a) Use this reursion formula, c_j+1 = (2(j+l+1-n)*c_j)/((j+1)(j+2l+2)), to confirm that when l=n-1 the radial wave function takes the form:
R_n,n-1 = (N_n)*r^(n-1)*e^(-r/(na))
(b) Calculate and for states psi_n,n-1,m.

(See attached file for full problem description)
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Use the following table to solve 3 and 4.
J0(x) J1(x) Y0(x) Y1(x)
2.4048 0.0000 0.8936 2.1971
5.5201 3.8317 3.9577 5.4297
8.6537 7.0156 7.0861 8.5960
11.7915 10.1735 10.2223 11.7492
14.9309 13.3237 13.3611 14.8974
3. Find the first four α i

Problem 1: Given the metric space (X, p), prove that
a) |p(x, z) - p(y, u)| < p(x, y) + p(z, u) (x, y, z, u is an element of X);
b) |p(x, y) - p(y, z)| < p(x, y) (x, y, z is an element of X).
These problems are from Metric Space. Please give formal proofs for both (a) and (b) based on the reference provided. Thank y

Please help with the following proofs. Answer true or false for each along with step by step proofs.
1) Prove that all integers a,b,p, with p>0 and q>0 that
((a+b) mod p)mod q = (a mod p) mod q + (b mod p) mod q
Or give a counterexample
2) prove for all integers a,b,p,q with p>0 and q>0 that
((a-b)mod p) mod q=0

7. This problem generalizes the factorial function, as in n!=n(n-1)(n-2)...(2)(1), to more general arguments than just the positive integers.
(a) Use integration by parts to show that for any positive integer n, the integral with respect to x from 0 to infinity of xne-x is n!
(b) Make a clear case that the integral exists

I need a detailed script in order to show this works in MATLAB.
This problem. All I need is the circled #3 problem, but only required to do the top a) and b) graphs.

In this problem, you will find the electrostatic potential inside an infinitely long, grounded, metal cylinder of unit radius whose axis coincides with the z-axis (See figure below). In cylindrical coordinates, the potential, V(r, theta, z), satisfies Laplace's equation... Please see attached... Let us assume that the po