
11. Solve this boundary value problem for u(x, t): n2 xu,-(x14),--11 (0<x <c,0 11 (c, 1) = 0, u(x, 0) = f(x), where u is continuous for0sxc,0 and where n is a positive integer. Answer: u(x, 1) Σ A,Jn(gjx) exp (-α,1), where a", and A, are the constants j-1
11. Solve this boundary value problem for u(x, t): n2 xu,-(x14),--11 (0
If ū +3+w -0, show that u u xe - w xu
Prove that each of the metrics (4.1) , (4.2) , and (4.3) has
property (4.4) .
211/2 4.2) (4.3) A sequence {xu) = (xe)-1 converges to x = (지,. . .x,) in X if and only if for each k the sequence of component entries {XkU)}-i converges to x, in Xk. (4.4)
211/2
4.2) (4.3)
A sequence {xu) = (xe)-1 converges to x = (지,. . .x,) in X if and only if for each k the sequence of...
Pro The more resource in the show is there are con that the best het vlees who the - Vand - Calculate the power (a) to the currenthe Show Work is REQUIRED for this questions on Show Work
If E [exp(aX)] exists for a given constant a, then show that for t > 0 (a) exp(−at)P (X > t) < E [exp(aX )] , if a > 0. (b) exp(−at)P (X < t) < E [exp(aX )] , if a < 0.
Let XU(a, b) be a uniformly distributed random variable. Use the definition of mean and variance to show that: (a) E(X)t (b) Var(x)2
a) Show that the wave function y(x) = N exp( – x²/(2a?)) with a? = () is a solution of the Schrödinger equation for harmonic oscillator with potential V(x) = k x2/2. (10 pt) b) What is the energy of harmonic oscillator with the wave function y(x) in terms of k and m? (5 pt) c) Sketch the potential energy of harmonic oscillator, the energy level corresponding to y(x), the wave function (x), and the probability density associated with y(x)...
1. Suppose that the joint density of X and Y is given by exp(-y) (1- exp(-x)), if 0 S y,0 syS oo exp(-x) (1- exp(-y)), if 0SyS ,0 oo (e,y)exp(-y) Then . The marginal density of X (and also that of Y), ·The conditional density of Y given X = x and vice versa, Cov(X, Y) . Are X and Y independent? Explain with proper justification.
4 Linearize the following ODE around Xo 2T,u,-1 0 0 x 2 sin(x) + xu + u2
(a) Show the expression of the Hamiltonian operator for the single electron in Het and Write out the exact wavefunction expression for an electron in He* in the quantum state of (3,1,0). (b) Determine the values for the angular momentum and energy for the electron in the quantum state of (3,1,0).