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Picard's method solution vs exact solution

  1. Mar 19, 2013 #1
    1. The problem statement, all variables and given/known data

    I have to use Picard's method to find an approximate solution to

    [itex]\frac{dy}{dx}=2xy[/itex]

    and then compare it to the exact solution, given that [itex]y(0)=2[/itex]

    2. Relevant equations

    Picard's method:

    [itex]y_{n}=y_{0} + ∫^{x}_{x_{0}} f(\phi,y_{n-1}(\phi))d\phi[/itex]

    3. The attempt at a solution

    So I go over Picard's method about 3 times and get the result:

    [itex]2(1+x^2 + \frac{x^4}{2} + \frac{x^6}{6})[/itex]

    which looks like it miiiight be going towards:

    [itex]2(\sum ^{\infty}_{n=0} \frac{x^{2n}}{n!}) = 2e^{x^2}[/itex]


    I then solve the DE exactly by noting that the DE is separable, so I separate and integrate to obtain:

    [itex]y= e^{x^2}+C[/itex]

    Using my initial condition, I find that C=1. So, my question is, have I done this correctly? I have a feeling that I'm supposed to get the same answer for both.
     
  2. jcsd
  3. Mar 19, 2013 #2

    Dick

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    e^(x^2)+1 is NOT a solution to the ODE. Try it. You messed up where the constant C belongs.
     
  4. Mar 19, 2013 #3
    Argh! Silly mistake. Corrected it and now they are equal, thanks :)
     
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