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Math Help - Extremals

  1. #1
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    Extremals

    Calculate and sketch the extremals of integral (from 1 to b) of (x^2*y'^2 + 12y^2) dx which pass through (1,1) and where b>1.


    There is a hint, to solve the Euler-Lagrange equation, try solutions of the form y = x^p but I don't see how that helps.


    Any help would be appreciated.

    Thanks.
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  2. #2
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    Well, what is the Euler-Lagrange equation for this problem?
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  3. #3
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    (x^2)y'' + 2xy' -12y = 0

    ???
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  4. #4
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    Quote Originally Posted by bobby View Post
    (x^2)y'' + 2xy' -12y = 0

    ???
    Okay, and the other hint was to try y= x^p. Then y'= px^{p-1} and y"= p(p-1)x^{p-2}. Putting those into the equation we have x^2(p(p-1)x^{p-2}+ 2x(px^{p-1})- 12x^p= [p(p-1)+ 2p- 12]x^p= 0 which will be true, for all x, if and only if p(p-1)+ 2p- 12= p^2+ p- 12= (p-3)(p+4)= 0 so the general solution to the Euler-LaGrange equation is y(x)= Cx^3+ Dx^{-4}. Now, what does the Euler-LaGrange equation tell you about the extremals?
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  5. #5
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    Thanks, so does that mean the coefficient of the first derivative is

    Cx^3+ Dx^{-4}

    ? Not sure on this.

    just wondering, did I get my ELeqn correct or is it meant to be 6y - xy' = 0 ? As I'm not sure if you're meant to differentiate both terms in d/dx ( 2x^2 * y') as y' is a function of x..?
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