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Boundary value problem(ODEs)

  1. Nov 29, 2009 #1
    Dear everybody,

    I am new here. I need your suggesstion and helping for solving the next nonlinear boundary value problem:

    x'=-y/sqrt(x^2+y^2)
    y'=x/sqrt(x^2+y^2)
    with x(0)=y(0)=-1 and x(pi/4)=y(pi/4)=1

    so is it easy to get the analytical solution to the problem above or I have to solve this problem numerically.
    thanks in advance

    Amr
     
    Last edited: Nov 29, 2009
  2. jcsd
  3. Nov 29, 2009 #2
    Using polar coordinates, and rewriting in vector form you get the equivalent equation:

    [tex]\frac{d\vec{r}}{dt}=\hat{\theta}[/tex] (1)




    [tex]\frac{d\vec{r}}{dt}=\hat{r}\frac{dr}{dt}+r\frac{d\theta}{dt}\hat{\theta}[/tex]

    Putting this in the equation (1) and comparing different components you get:

    [tex]\frac{dr}{dt}=0 => r(t) \equiv const=R[/tex]
    [tex]r\frac{d\theta}{dt}=1 [/tex]

    Since r is constant you get:

    [tex] \theta=\frac{t}{r}+\varphi [/tex]

    From this you get that:

    [tex] x(t)=Rcos(\frac{t}{R}+\varphi); y(t)=Rsin(\frac{t}{R}+\varphi); [/tex]

    You have a circular motion with a radius and initial phase to determine from boundry conditions.
     
    Last edited: Nov 30, 2009
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