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Existence of solution legendre equation

  1. Aug 23, 2013 #1
    Hi all, I have my exam in differential equations in one week so I will probably post alot of question. I hope you wont get tired of me!


    1. The problem statement, all variables and given/known data
    This is Legendres differential equation of order n. Determine an interval [0 t_0] such that the basic existence theorem guarantees a solution.

    (1-t^2) [itex]\frac{d^2y}{dt^2}[/itex] - 2t[itex]\frac{dy}{dt}[/itex]+n(n-1)y = 0
    y(0)=0
    y'(0)=0


    2. Relevant equations

    Picard-Lindelöfs theorem
    M = max |f(x)|

    T_0 = min {T,δ/M}

    3. The attempt at a solution

    I thought, Picards theorem is for first order equation so thought I should first rewrite the system as x'(t)=A(t)x(t)

    Write

    [itex]x_{1}[/itex] = y --- >[itex]dx_{1}/dt[/itex] = dy/dt

    [itex]x_{2}[/itex] = dy/dt --- > [itex]dx_{2}/dt[/itex] = d^2y/dt^2

    dx_1/dt = [itex]\frac{1}{1+t^2}[/itex]

    dx_2/dt = [itex]\frac{-n(n+1)}{1+t^2}[/itex] + [itex]\frac{2t)}{1+t^2}[/itex]


    Picards theorem require the function to be lipschitz, which it seems to be.

    Picards theorem also states that you should pick the interval according to

    M = max |f(x)|

    T_0 = min {T,δ/M}

    Here is where I am stuck, assuming I correct so far.

    M = max |f(x)|

    T_0 = min {T,δ/M}

    I think M is equal to n(n+1) but I have NO idea on how to compute T_0.


    As always, all help is greatly appreciated!
     
  2. jcsd
  3. Aug 26, 2013 #2
    Some corrections to my current solution
     
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