Homework Statement
Solve the problem.
utt = uxx 0 < x < 1, t > 0
u(x,0) = x, ut(x,0) = x(1-x), u(0,t) = 0, u(1,t) = 1
Homework Equations
The Attempt at a Solution
Here is what I have so far but I'm not sure if I am on the right path or not.
u(x,t) = X(x)T(t)...
Homework Statement
I need to visualize the wave equation with the following initial conditions:
u(x,0) = -4 + x 4<= x <= 5
6 - x 5 <= x <= 6
0 elsewhere
du/dt(x,0) = 0
subject to the following boundary conditions:
u|x=0 = 0
Homework Equations
I'm not sure I understand the...
Homework Statement Derive the differential equation governing the longitudinal vibration of a thin cone which has uniform density p, show that it is
1/x/SUP] d/dx(x du/dx) = (1/c) d u/d[SUP]t
Hint: The tensile force sigma = E du/dx where E is the Young's modulus (a constant), u is the...
Homework Statement
Show that the solution u(r,theta) of Laplace's equation (nabla^2)*u=0 in the semi-circular region r<a, 0<theta<pi, which vanishes on theta=0 and takes the constant value A on theta=pi and on the curved boundary r=a, is
u(r,theta)=(A/pi)[theta + 2*summation ((r/a)^n*((sin...
Homework Statement
Obtain all solutions of the equation partial ^2 u/partial x^2 - partial u/partial y = u of the form u(x,y)=(A cos alpha x + B sin alphax)f(y) where A, B and alpha are constants. Find a solution of the equation for which u=0 when x=0; u=0 when x = pi, u=x when y=1...
Homework Statement
Show that u=f(2x+y^2)+g(2x-y^2) satisfies the equation y^2 d^2u/dx^2 + (1/y) du/dy - d^2u/dy^2=0 where f and g are arbitrary (twice differentiable) functions.
Homework Equations
The Attempt at a Solution
I came up with fxx=0 fyy=2 gxx=0 gyy= 2. But didn't...
Homework Statement
I am trying to do our the review exam our teacher posted to study for a test and I am having difficulty trying to figure out where to start and what to do. Our teacher lost me when he was explaining this section. Please help!
Homework Equations
Write the power...
Homework Statement
Find the limit for the following sequence and then use the definition of limit to justify your result.
Homework Equations
n^3/(2*n^3 + n)
The Attempt at a Solution
I found the limit as n --> infinity is 1/2. I think the next step is to set up the equation as...