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Writing a system of 2 ODEs as a 1st order ODE

  1. Feb 3, 2012 #1
    1. The problem statement, all variables and given/known data

    Consider the following initial value problem for two functions [itex]y(x),z(x)[/itex]: [tex]0 = y''+(y'+7y)\text{arctan}(z)[/tex] [tex]5z' = x^2+y^2+z^2[/tex] where [itex]0 \leqslant x \leqslant 2,\; y(0)=1.8,\;y'(0)=-2.6,\;z(0)=0.7[/itex].

    Rewrite the system of ODEs in standard form using a suitable substitution.

    3. The attempt at a solution

    Would putting [itex]{\bf u} = \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix}[/itex] where [itex]u_1(x) = y(x),\; u_2(x) = y'(x),\; u_3(x)=z(x)[/itex] work?


    [tex]u_1' = y' = u_2[/tex] [tex]u_2' = y'' = -(y'+7y)\arctan(z) = -(u_2+7u_1)\arctan(u_3)[/tex] [tex]u_3' = z = \frac{1}{5} ( x^2 + u_1^2 + u_3^2)[/tex]

    so that [itex]{\bf u'} = \begin{bmatrix} u_1' \\ u_2' \\ u_3' \end{bmatrix} = \begin{bmatrix} u_2 \\ -(u_2+7u_1)\arctan(u_3) \\ \frac{1}{5} ( x^2 + u_1^2 + u_3^2) \end{bmatrix} ,\; 0 \leqslant x \leqslant 2[/itex] and [itex]{\bf u}(0) = \begin{bmatrix} 1.8 \\ -2.6 \\ 0.7 \end{bmatrix}[/itex]

    Last edited: Feb 3, 2012
  2. jcsd
  3. Feb 3, 2012 #2


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    Science Advisor

    Yes, that is perfectly correct. Note that this does NOT reflect your title! You are NOT "Writing a system of 2 ODEs as a 1st order ODE". You are, rather, writing a higher order system of equations as a system of first order ODEs.
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