Results 1 to 2 of 2

Thread: how to show this DE has infinitely many eigenvalues

  1. #1
    Senior Member
    Joined
    Feb 2008
    Posts
    321

    how to show this DE has infinitely many eigenvalues

    $\displaystyle y''+\lambda y=0 \ \ \ \ \ \ y(0)=0, \ \ y(1)+y'(1)=0 $

    has infinitely many eigenvalues $\displaystyle \lambda_{1}<\lambda_{2}<... $

    and indicate the behaviour of lambda as n goes to infinity.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    10
    You really ought not to need much of a hint in order to do this. You should recognise $\displaystyle y''+\lambda y=0$ as a simple harmonic motion equation (assuming that $\displaystyle \lambda>0$), with solution $\displaystyle y=A\cos\mu x + B\sin\mu x$, where $\displaystyle \mu = \pm\sqrt\lambda$. The initial conditions will give you a condition on $\displaystyle \mu$, of the form $\displaystyle \tan\mu=-\mu$. This is not an equation that you can solve explicitly, but by drawing a graph of the tan function, you can see that there is a doubly-infinite family of solutions, which for large values of $\displaystyle |\mu|$ will be close to odd multiples of $\displaystyle \pi/2$.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Show that every open interval has infinitely many points
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: Jun 4th 2011, 09:26 AM
  2. Show that Z[(3)^(1/2)] has infinitely many units
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Jan 21st 2011, 05:06 AM
  3. Infinitely many primes
    Posted in the Number Theory Forum
    Replies: 11
    Last Post: Jun 4th 2010, 05:43 AM
  4. how to show the eigenvalues of a jacobi matrix.
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Nov 29th 2009, 12:48 AM
  5. Infinitely or no solution?
    Posted in the Algebra Forum
    Replies: 2
    Last Post: Sep 25th 2007, 03:24 PM

Search Tags


/mathhelpforum @mathhelpforum