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Calculate the first four Picard Iterates of the equation y' - y = x^2

  1. Apr 10, 2007 #1
    Help Please!!!!

    Calculate the first four Picard Iterates of the equation y' - y = x^2 with the condition y(0) = -1

    and it was given that y'(x) = x^2 +y and y(0)= -1



    Need a little help with this question... Not sure what to do.
     
  2. jcsd
  3. Apr 11, 2007 #2
    The definition of Picard iterates is

    [tex]\phi_0=y(x_0),[/tex]

    [tex]\phi_{n+1}(x)=\phi_0+\int_{x_0}^x f(s,\phi_n(s))ds,[/tex]

    where [itex]f[/itex] is given by

    [tex]y'(x)=f(x,y(x)).[/itex]

    In your case, [itex]f(x,y(x))=x^2+y[/itex], [itex]\phi_0=y(0)=-1[/itex], and

    [tex]\phi_1=-1+\int_0^x (s^2+\phi_0)ds=-1+\int_0^x (s^2-1)ds.[/tex]

    What is [itex]\phi_2[/itex]? and [itex]\phi_n[/itex]?
     
    Last edited: Apr 11, 2007
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