Results 1 to 2 of 2

Math Help - Linear second order, homogeneous, constant coefficients, unchanged by all rotations

  1. #1
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    Linear second order, homogeneous, constant coefficients, unchanged by all rotations

    Show that the only second-order linear homogeneous equation with constant coefficients in x and y whose form is unchanged by all rotations of axes is u_{xx}+u_{yy}+ku=0.

    \omega_{\xi\xi}(\cos^2+\sin^2)+\omega_{\eta\eta}(\  cos^2+\sin^2)+k\omega=0\Rightarrow\omega_{\xi\xi}+  \omega_{\eta\eta}+k\omega=0\Rightarrow u_{xx}+u_{yy}+ku=0

    Showing this doesn't change is the easy part (see above), but how do I show it is the only second-order linear?

    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,392
    Thanks
    56
    Quote Originally Posted by dwsmith View Post
    Show that the only second-order linear homogeneous equation with constant coefficients in x and y whose form is unchanged by all rotations of axes is u_{xx}+u_{yy}+ku=0.

    \omega_{\xi\xi}(\cos^2+\sin^2)+\omega_{\eta\eta}(\  cos^2+\sin^2)+k\omega=0\Rightarrow\omega_{\xi\xi}+  \omega_{\eta\eta}+k\omega=0\Rightarrow u_{xx}+u_{yy}+ku=0
    Showing this doesn't change is the easy part (see above), but how do I show it is the only second-order linear?

    Thanks.
    Start with

    a u_{xx} + b u_{xy} + c u_{yy} + du_x + e u_y + f u = 0.

    Introduce your change of variables and require that

    a \omega_{\xi \xi} + b \omega_{\xi \eta} + c \omega_{\eta \eta} + d \omega_{\xi} + e \omega_{\eta} + f u = 0.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 4
    Last Post: May 21st 2011, 03:01 PM
  2. Non-homogeneous 2nd order DE with constant coefficients.
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: April 8th 2010, 01:15 PM
  3. Second-Order Linear Non Homogeneous ODE with Constant Coefficients
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: October 23rd 2009, 12:13 AM
  4. second order differential constant coefficients homogeneous
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: October 1st 2008, 10:50 AM
  5. Replies: 1
    Last Post: July 29th 2007, 03:37 PM

/mathhelpforum @mathhelpforum