Question

Verify that the function in the following question is the
solution of wave equation

w=f(u), where f is a differentiable function of u, and u =
a(x+ct), where a is a constant

Answer #1

Verify that u(x, t) = v(x + ct) + w(x − ct) satisfies the wave
equation for any twice differentiable functions v and w.

Verify by substitution that U(x,t) = A eB(x-vt) is a
solution to the wave equation when A and B are constants.

Determine whether each of the following functions is a solution
of wave equation: a) u(x, t) = sin (x − at), b) u(x, t) = sin (x −
at) + ln (x + at)

The state of a particle is completely described by its
wave function Ψ(?,?) One-dimensional Schrodinger Equation-- answer
the following questions:
2) Show that when U(x) = 0, and , is a solution to the
one-?=2??/ℏΨ=?sin??dimensional Schrodinger equation.
3) Show that when U(x) = 0, and , is a solution to the
one-?=2??/ℏΨ=?cos??dimensional Schrodinger equation.
4) Show that where A and B are constants is a solution
to the Ψ=??+?Schrodinger equation when U(x) = 0, and when E =
0.

Show that x(t) = A sin(wt) sin(kx) satisfies the wave equation,
where w and k are some constants. Find the relation between w, k,
and v so that the wave equation is satisfied.

The wave function of a wave is given by the equation
D(x,t)=(0.2m)sin(2.0x−4.0t+π),
where x is in metres and t is in seconds.
a. What is the phase constant of the wave?
b. What is the phase of the wave at t=1.0s and x=0.5m?
c. At a given instant, what is the phase difference between two
points that are 0.5m apart?
d. At what speed does a crest of the wave move?

Consider an individual whose utility function over wealth is
U(W), where U is increasing smoothly in W (U’ > 0) and convex
(U’’ > 0).
a. Draw a utility function in U-W space that fits this
description.
b. Explain the connection between U’’ and risk aversion
c. True or false: this individual prefers no insurance to an
actuarially fair, full contract. Briefly explain your answer

For the function w=f(x,y) , x=g(u,v) , and
y=h(u,v). Use the Chain Rule to
Find ∂w/∂u and
∂w/∂v when u=2 and v=3 if
g(2,3)=4, h(2,3)=-2,
gu(2,3)=-5,
gv(2,3)=-1 ,
hu(2,3)=3,
hv(2,3)=-5,
fx(4,-2)=-4, and
fy(4,-2)=7
∂w/∂u=
∂w/∂v =

Verify that the indicated function is a solution of the given
dierentail
equation. In some cases assume an appropriate interval of validity
for the
solution.
y''+y'-12y=0; y=c1e^3x+c2e^-4x

Show that the function f(x, t) = x 2 + 4axt−4a 2 t 2 satisfies
the wave equation if one assumes a certain relationship between the
constant a and the wave speed u. What is this relationship?

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