Search results

  1. W

    PDE Separation of variables

    Sorry to be so dense, but I get lost at this point. I think I am then suppose to do ut=X(x)[-C(∏/2 + 2m∏)sin(∏/2 + 2m∏)t + D(∏/2 + 2m∏)cos(∏/2 + 2m∏)t) ut(x,0) = D(∏/2 + 2m∏)cos(∏/2 + 2m∏) = x(1-x) ----> D ≠ 0 t = 0 f(x) = ∑ Dsin (∏/2 + 2m∏)t u(x,t) = ∑ D sin ((∏/2 + 2m∏)t sin...
  2. W

    PDE Separation of variables

    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)...
  3. W

    PDE Wave Equation/boundary condition question

    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...
  4. W

    PDE - Need help getting started.

    Thank you for your patience and help in explaining this in a manner that I was able to understand.
  5. W

    PDE - Need help getting started.

    Okay now I have the following: F = ma F = EA(x)U(x) = d/dx[EA(x)d/dx)] = m(x) d^2u/dt^2 u(x,t) = U(x)T(t) d/dx[EA(x) d/dx[U(x)T(t)]] = d^2/dt^2 [m(x)T(t)] t(t) d/dx [EA(x) d/dx [U(x)T(t)] = 1/T(t) d^2/dt^2 T(t) = -w^2 d/dx[EA(x) d/dx U(x)] + w^2m(x)U(x) = 0 I need to get to 1/x^2 d/dx[x^2...
  6. W

    PDE - Need help getting started.

    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...
  7. W

    PDE Cylindrical and Spherical Symmetry

    okay I have been studying my book all weekend and this is what I had. Laplace Equation (1/r) d/dr(rdu/dr) + (1/r^2)d^2u/ds = 0 (s represents theta) u(r,s)=R(r)S(s) (1/Rr)d/dr(rdR/dr) + (1/(r^2S))(d^2S/ds^2) = 0 (1/S)d^2S/ds^2 = -w^2 and (r/R)d/dr(rdR/dr)= w^2 Therefore S = A cos ws +...
  8. W

    PDE Cylindrical and Spherical Symmetry

    Yes I do understand and know the method of Separation of variables.
  9. W

    PDE Cylindrical and Spherical Symmetry

    This was all the data I was given. Our book is very brief and assumes you are very proficient with ODE's. I am struggling and need to understand these concepts. Thanks!
  10. W

    PDE Cylindrical and Spherical Symmetry

    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...
  11. W

    PDE - Fourier Method

    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...
  12. W

    Partial Differential Equations Help!

    Sorry to be a pain but I think I am finally getting the idea and want to make sure this is true. ux = f'(2x+y2)*2 + g'(2x-y2)*2 uy = f'(2x+y2)*2y + g'(2x-y2)*-2y uxx = f''(2x+y2)*0 + g''(2x-y2)*0 uyy = f''(2x+y2)*2 + g''(2x-y2)*-2 then I just plug these results into the equation y2uxx +...
  13. W

    Partial Differential Equations Help!

    I am still confused, I do not see where the chain rule comes into play, f(x,y) = 2x + y2. So why isn't fx equal to 2? I do not see how you are coming up with (2x + y2)*2. I desperately need to understand this whole concept! Thanks!
  14. W

    Partial Differential Equations Help!

    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...
  15. W

    Power expansion

    Please accept my apologies. I had been studying all day and had hit a wall. I stepped away and when I came back, it all made sense. Your assistance was greatly appreciated! Thank you for all your help!
  16. W

    Power expansion

    Sorry to be so dense but I really have hit a wall where this concept is concerned. I understand what you are saying but not sure what you meant to do next. I really need to understand this concept as I have a test coming up! Thanks!
  17. W

    Power expansion

    Thanks! But that is not what he is looking for. I need to come up with a summation. One example he worked for us in class was e^(-x^2) = summation (-1)^n [(x^(2n))/n!].
  18. W

    Power expansion

    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...
  19. W

    Find the limit of the sequence n^3/(2n^3 + n)

    okay, I came up with 1/(2+1/n^2) which is where I came up the idea that the limit as n---->infinity is 1/2. But then if I set it to be < epsilon. How do you know what epsilon should be? I am really struggling with this concept. I appreciate your help and patience.
  20. W

    Find the limit of the sequence n^3/(2n^3 + n)

    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...
Top