Results 1 to 2 of 2

Thread: Diff Eq Translated Matrix?

  1. #1
    Mar 2014

    Diff Eq Translated Matrix?

    This is probably a trivial problem, considering I have done most of the work already:


    This system is almost linear with Crit Points at 0,0 and 3/8,3/2

    My problem is that I cannot figure out for the life of me what the "translated matrix" of the point is 3/8,3/2 is in order to classify it. Does anyone know how to do this?

    Thank you in advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Apr 2005

    Re: Diff Eq Translated Matrix?

    First do the translation: you want to "move" x and y so that x= 3/8, y= 3/2 is moved to 0, 0. Do that by taking u= x- 3/8 and v= y- 3/2. Then x= u+ 3/8, y= v+ 3/2, x'= u, y'= v', and $\displaystyle y^2= (v+ 3/2)^2= v^2+ 3v+ 9/4$ so the equations become $\displaystyle x'= u'= -2(u+ 3/8)- (v+ 3/2)+ v^2+ 3v+ 9/2= -2u+ 2v+ v^2$ and [tex]y'= v'= -4(u+ 3/8)+ (v+ 3/2)= -4u+ v.

    Now there are two, equivalent and very similar, ways to continue. The first is to recognize that the first equation is non-linear because of the "$\displaystyle v^2$". The only way to "linearize" that is to drop the $\displaystyle y^2$: u'= -2u+ 2v and v'= -4u+ v. The "translated matrix" is $\displaystyle \begin{bmatrix}-2 & 2 \\ -4 & 1\end{bmatrix}$.

    The other way is to form the "Jacobian". If $\displaystyle x'= f(x,y)$ and $\displaystyle y'= g(x,y)$ the Jacobian is $\displaystyle \begin{bmatrix}\frac{\partial f}{\partial x} & \frac{\partial g}{\partial x} \\ \frac{\partial f}{\partial y} & \frac{\partial g}{\partial y}\end{bmatrix}$

    In this case that would be [tex]\begin{bmatrix}$\displaystyle \begin{bmatrix}-2 & 2+ 2v \\ -4 & 1\end{bmatrix}$. Now let u and v be 0 which gives the same matrix as before. This second method is more "formal" and helps determine how to "linearize" a formula.
    Last edited by HallsofIvy; Mar 6th 2014 at 07:29 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 7
    Last Post: Aug 5th 2011, 01:30 PM
  2. Describe region of translated sphere
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Nov 17th 2010, 01:37 AM
  3. Fourier transform of a translated variable
    Posted in the Calculus Forum
    Replies: 0
    Last Post: Aug 2nd 2010, 01:41 AM
  4. diff eq's
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Mar 27th 2009, 07:45 PM
  5. Replies: 1
    Last Post: Oct 9th 2007, 11:30 PM

Search Tags

/mathhelpforum @mathhelpforum