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Back Euler method for 2nd order d.e

  1. Oct 12, 2012 #1
    Hi,
    How can one use back Euler method for 2nd order d.e?

    Is it possible this method to be expanded for a system of 4 odes?

    Thanks
     
  2. jcsd
  3. Oct 12, 2012 #2

    AlephZero

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    Write ##dy/dx = p## and ##d^2y/dx^2 = dp/dx##, then solve two first order equations for ##y## and ##p##. (The first equation is your original DE rewritten using ##y##, ##p##, and ##dp/dx##. The second equation is just ##dy/dx = p##).

    Yes. Just write the 4 variables as a vector, so your equiations become one equation like $$\frac{d}{dx}\begin{pmatrix}y_1 \\ y_2 \\ y_3 \\ y_4\end{pmatrix} = f\begin{pmatrix}y_1 \\ y_2 \\ y_3 \\ y_4\end{pmatrix} + g(x)$$
     
  4. Oct 12, 2012 #3
    Ok.. and then how the back Euler scheme will be like..?
    If it was one ode it would be: [itex]y_{n+1}[/itex]=[itex]y_{n}[/itex]+f( [itex]y_{n+1}[/itex], [itex]t_{n+1}[/itex]).
    To be more specific I want to solve the system:[itex]y^{4}[/itex]=1/[itex]y^{2}[/itex], with y(0)=0,y''(0)=0, applying this method..
    Thanks.
     
  5. Oct 12, 2012 #4

    AlephZero

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    It looks exactly the same. Just replace the scalar y with the vector.
     
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