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Find extremal

  1. Feb 19, 2014 #1
    1. The problem statement, all variables and given/known data
    Find extremals of the functional
    ##\Phi(y,z)=\int^{\frac{\pi}{2}}_0((y')^2+(z')^2+2yz)dx##
    for
    ##y(0)=0##, ##y(\frac{\pi}{2})=1##, ##z(0)=0##, ##z(\frac{\pi}{2})=-1##


    2. Relevant equations



    3. The attempt at a solution
    Well I have a solution but I have problem how to start with it
    Solution
    System of equation
    ##y''-z=0##
    ##z''-y=0##
    Differentiating first equation two times and add second equation two given result we obtain
    ##y^{(4)}-y=0##
    ##y=C_1e^x+C_2e^{-x}-C_3\cos x+C_4\sin x##
    And from boundary condition
    ##C_1=C_2=C_3=0##
    ##C_4=1##
    We obtain extremals
    ##y=\sin x## and ##z=-\sin x##
    My problem is how to get this system of equations. Tnx for your help.
     
  2. jcsd
  3. Feb 19, 2014 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    You just use the Euler-Lagrange equation to derive that system, LagrangeEuler. Can you state what that is to get started anyway?
     
  4. Feb 19, 2014 #3
    Tnx. I solved the problem.
     
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