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Math Help - Picard iter

  1. #1
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    Question Picard iter

    u'_1 = x*y_1 - u_2 , u_1(o)=1

    u'_2 = xu^2_1 - u_2, u_2(o)=1

    Using picard iteration, show that the soln can be obtained for |x|<o.3


    Do you find u_1 and u_2 through picard iteration first?
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  2. #2
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    Quote Originally Posted by Nusc View Post
    u'_1 = x*y_1 - u_2 , u_1(o)=1

    u'_2 = xu^2_1 - u_2, u_2(o)=1

    Using picard iteration, show that the soln can be obtained for |x|<o.3


    Do you find u_1 and u_2 through picard iteration first?
    Recall that if \frac{dy}{dx} = f(x,y),\;\; y(a) = b then the picard iterates are obtained by solving

    y_{n+1} (x) = b + \int_a^x f(s,y_n(s))\,ds<br />
,<br />
,y_0 = b

    For systems, this extends and for your system (I'll use x(t) and y(t) b/c of the subscripts)

    x_{n+1}(t) = 1 + \int_0^t s x_n(s) - y_n(s)\, ds<br />

    y_{n+1}(t) = 1 + \int_0^t s x_n^2(s) - y_n(s)\, ds

    with x_0 = 1,\;\;y_0 = 1. Then calculate the iterates recursively.
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