Results 1 to 11 of 11

Math Help - Reduce the differential equation to first order

  1. #1
    Super Member
    Joined
    Jun 2008
    Posts
    829

    Reduce the differential equation to first order

    How reduce the differential equation to first order ?

    For example:
    \frac{d^5y}{dx^5}+c_1 \frac{d^4y}{dx^4}+c_2 \frac{d^3y}{dx^3}+c_3 \frac{d^2y}{dx^2}+c_4 \frac{dy}{dx} = f(x)

    and write the resulting system of equations in the form:
    \vec{z} = A \vec{z} + Bf(x)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    Wrong forum, I think. Anyway. What ideas have you had so far?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Jun 2008
    Posts
    829
    Quote Originally Posted by Ackbeet View Post
    Wrong forum, I think. Anyway. What ideas have you had so far?
    That's the problem. I do not know the formula to reduce to the first order
    Follow Math Help Forum on Facebook and Google+

  4. #4
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    See here for a very similar problem. Note that in your case, the function itself is not present.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Jun 2008
    Posts
    829
    Quote Originally Posted by Ackbeet View Post
    See here for a very similar problem. Note that in your case, the function itself is not present.
    I will calculate at hand.
    That link you now, I did not get any formula to reduce the differential equation
    Follow Math Help Forum on Facebook and Google+

  6. #6
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    It's not really a formula. It's a procedure. You let higher-order derivatives be components of the vector z. Here's another link.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,313
    Thanks
    1291
    \vec{z}= \begin{pmatrix}y \\ \frac{dy}{dx} \\ \frac{d^2y}{d^2x} \\\frac{d^3y}{dx^3}\\ \frac{d^4y}{dx^4}\end{pmatrix}
    Follow Math Help Forum on Facebook and Google+

  8. #8
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    You actually don't need the first component of y there, since there is no y in the original DE. Only the derivatives show up. So you can get by with a 4 x 4 system.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,313
    Thanks
    1291
    Right. If you write u= \frac{dy}{dx} then the differential equation can be written
    \frac{d^4u}{dx^4}+c_1 \frac{d^3u}{dx^3}+c_2 \frac{d^2u}{dx^2}+c_3 \frac{du}{dx}+c_4 u = f(x)
    However, since that was given as a "for example", I suspect the y was inadvertantly left out.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Super Member
    Joined
    Jun 2008
    Posts
    829
    Quote Originally Posted by HallsofIvy View Post
    Right. If you write u= \frac{dy}{dx} then the differential equation can be written
    \frac{d^4u}{dx^4}+c_1 \frac{d^3u}{dx^3}+c_2 \frac{d^2u}{dx^2}+c_3 \frac{du}{dx}+c_4 u = f(x)
    However, since that was given as a "for example", I suspect the y was inadvertantly left out.
    Sorry, but I do not understand. Are not in the order 4 ?
    Follow Math Help Forum on Facebook and Google+

  11. #11
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    Technically, your original DE is order 5. However, since you left out the y term (the one with no derivative), you can use the substitution HallsofIvy mentioned, and immediately get a 4th order DE in u. Then, once you've found u, integrate once more to get y.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: September 21st 2011, 03:37 PM
  2. 4th order differential equation??
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: April 22nd 2010, 09:08 PM
  3. First Order Differential Equation
    Posted in the Differential Equations Forum
    Replies: 5
    Last Post: February 18th 2009, 09:20 AM
  4. Replies: 2
    Last Post: November 25th 2008, 09:29 PM
  5. second order differential equation help
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 2nd 2008, 07:11 AM

Search Tags


/mathhelpforum @mathhelpforum