Patrick

Use the reduction of order formula

y_2 = y_1 int( (1/(y_1)^2) * e^-(int( p)

(where p is the coefficient of y

0 when the equation is in its normal form) to find a

second, linearly independent solution to the equation in (a).

So I've got y_1 = x^3 and p=5x

So after integrating 5x we're left with

y_2 = x^3 int( (1/(x^3)^2) * e^-((5x^2)/2)

Its here where I've gotten a bit stuck.

Evaluate the following complex integral

[; \[\oint_{C} \frac{z^2}{z^2 +4}\] ;]

where C is in turn the following circles

(i) [; C_1 = |z-i| = 2 ;]

(ii) [; C_2 = |z+2i| = 1 ;]

I've gone about getting the roots for z^2 +4 and then the limits given that the roots were 2i and -2i.

The limit for 2i was -2i and for -2i it was -2i/3.

Evaluate the following real integral

∫(from 0 to 2pi) cosθ/(3+sinθ) dθ

1) Your classmate challenges you with the following argument: If a particle is in an eigenstate of a one-dimensional box of width L, then we know its energy exactly. We also know that the energy in the box is purely kinetic. Hence we know the particle’s momentum exactly as well. This contradicts the Heisenberg uncertainty relation since the uncertainty in the particle position is finite (Δx<L). Show where your friend’s argument is wrong and why.

2) Consider the wave function: ψ(x,t)=Ae^((-λ|x|) ) e^(-iωt)

where A, λ, and ω are real, positive constants.

(a) Normalize Ψ.

Hint: Use the fact that the wavefunction is even (i.e., based on the absolute value of x and the minus sign in the exponent) and look up the definite integral.

Show that the solution is: A=√λ

(b) Compute the expectation values <x> and <x^2> .

Show (and justify) that the solutions are:

<x>=0

and

<x^2> = 1/(2λ^2)

All dice games are based on the discrete stable positional equilibria of a cube on a plane surface. We may think of the numbers showing on top of a dice as a quantized observable a.

i) What are the eigenvalues a_n of the observable for a single die?

ii) What is <a> for a "honest" dice thrown upon a flat table?

iii) One die is dropped on a table and observed to be in a sate a_n=3. The table is then shaken such that there s a probability of 1/3 that the die will roll off the original face onto one of the adjacent faces.

Write the state function of the die after the table is shaken.

Use the First Shift Theorem (L−1{F(s − a)} = e^(at) f(t), a constant) to find

the inverse Laplace transform of: (s+1)/(s^2 -4s)

Q:Use triple integration to find the volume of the region enclosed by z= x^2 + y^2 and z= 18 -x^2 -y^2

As far as I can tell its a sphere intersecting with a parabloid.

Let

A = A1(x, y, z)i + A2(x, y, z)j + A3(x, y, z)k

be a vector field and φ = φ(x, y, z) a scalar field. Prove the following two vector

calculus identities:

(a)

∇×(φA) = φ(∇ × A) + (∇φ)×A

(b)

∇·(φA) = (∇φ)·A + φ(∇ · A)

v1=yi-xj v2=xi+yj

(A) Prove that v1 is not conservative by showing that it is not possible to find a scalar function φ1(x, y, z) such that v1 = −∇φ1. Prove that v2 is conservative by showing that it is possible to find a scalar function φ2(x, y, z) such that v2 = −∇φ2.

(B) Prove that v1 is not conservative by showing that ∇ × v1 6= 0. Prove that v2 is conservative by showing that ∇ × v2 = 0