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Math Help - Differential equation

  1. #1
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    Exclamation Differential equation

    I must show that the polynomial  H_{n} satisfies a diferential equation.I used some notation(1,2..) so that I don't rewrite the same thing again. By diferentiating H_{n}(x) = -xH_{n-1}(x) - (n - 1)H_{n-2}(x) (1)and
    using induction on n,I'm supposed to show that, for n >= 1,
    H'_n(x) = -nH_{n-1}(x) (2)
    I have to use (2) to express  H_{n-1} and  H_{n-2} in terms of derivatives of  H_{n}, and substitute these into (1) to show that
    H''_{n} - xH'_{n} + nH_{n} = 0 (3)
    for n>= 0. Now let  \phi'_n(x) = exp(-(x^2)/4 )H_{n}(x). Using (3) I must show that
    \phi'_n(x) +(n+1/2-(x^2)/4)\phi_n(x)=0

    Also H_n is a polynomial defined as follows:
    H_{n}(x) for n = 0,1,2.... :  H_{0}(x) =1 and H_{1}(x) = -x ; then, for n >=2, H_{n} is defined by the recurrence
    H_{n}(x) = -xH_{n-1}(x) - (n - 1)H_{n-2}(x): (1)

    Attempt at a solution

    I need help with the entire problem mainly because for example H'_n(x) = -nH_{n-1}(x) I can only prove it for the base case when but fot the general I get H'_n(x) = -H_{n-1}(x)-xH_{n-1}(x)-(n - 1)H_{n-2}(x)=-H_{n-1}(x)-xH'_{n-1}(x)-nH'_{n-2}(x)-H'_{n-2}(x)= -H_{n-1}(x)-xnH'_{n-1}(x)+xH'_{n-2}(x)-n^2H'_{n-3}(x)+2nH'_{n-3}(x)-nH'_{n-3}(x)+2H'_{n-3}(x)......and I do not know what do do next...if it isn't obvious I just substituted in the relation that I have to prove since I'm assuming that everything from n-1 is true......
    Also for the other parts I still need help in showing (3) and (4).Thanks in advance
    Last edited by StefanM; February 18th 2011 at 11:54 AM.
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  2. #2
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    Quote Originally Posted by StefanM View Post
    I must show that the polynomial  H_{n} satisfies a diferential equation.I used some notation(1,2..) so that I don't rewrite the same thing again. By diferentiating H_{n}(x) = -xH_{n-1}(x) - (n - 1)H_{n-2}(x) (1)and
    using induction on n,I'm supposed to show that, for n >= 1,
    H'_n(x) = -nH_{n-1}(x) (2)
    I have to use (2) to express  H_{n-1} and  H_{n-2} in terms of derivatives of  H_{n}, and substitute these into (1) to show that
    H''_{n} - xH'_{n} + nH_{n} = 0 (3)
    for n>= 0. Now let  \phi'_n(x) = exp(-(x^2)/4 )H_{n}(x). Using (3) I must show that
    \phi'_n(x) +(n+1/2-(x^2)/4)\phi_n(x)=0

    Also H_n is a polynomial defined as follows:
    H_{n}(x) for n = 0,1,2.... :  H_{0}(x) =1 and H_{1}(x) = -x ; then, for n >=2, H_{n} is defined by the recurrence
    H_{n}(x) = -xH_{n-1}(x) - (n - 1)H_{n-2}(x): (1)

    Attempt at a solution

    I need help with the entire problem mainly because for example H'_n(x) = -nH_{n-1}(x) I can only prove it for the base case when but fot the general I get H'_n(x) = -H_{n-1}(x)-xH_{n-1}(x)-(n - 1)H_{n-2}(x)=-H_{n-1}(x)-xH'_{n-1}(x)-nH'_{n-2}(x)-H'_{n-2}(x)= -H_{n-1}(x)-xnH'_{n-1}(x)+xH'_{n-2}(x)-n^2H'_{n-3}(x)+2nH'_{n-3}(x)-nH'_{n-3}(x)+2H'_{n-3}(x)......and I do not know what do do next...if it isn't obvious I just substituted in the relation that I have to prove since I'm assuming that everything from n-1 is true......
    Also for the other parts I still need help in showing (3) and (4).Thanks in advance
    I'm having a really hard time reading your post, but I think you are talking about Hermite polynomials.

    Think link my be helpful.
    Hermite polynomials - Wikipedia, the free encyclopedia
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