1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Systems of First Order Linear Equations

  1. Nov 28, 2011 #1
    1. The problem statement, all variables and given/known data

    Systems of first order equations can sometimes be transformed into a single equation of
    higher order. Consider the system

    (1) x1' = -2x1 + x2
    (2) x2' = x1 - 2x2

    Solve the first equation for x2 and substitute into the second equation, thereby obtaining a second order equation for x1. Solve this equation for x1 and then determine x2 also.


    2. Relevant equations

    3. The attempt at a solution

    Solving (1) for x2 yields:

    x2 = x1' + 2x1

    Substituting into (2) yields:

    x2' = x1 - 2(x1' + 2x1)

    Simplifying...

    x2' = -3x1 - 2x1'

    How am I supposed to solve this now with the x2' term there?
     
  2. jcsd
  3. Nov 28, 2011 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    You aren't. You haven't yet reduced it to a higher order equation in one variable.
    "Solve the first equation for x2 and substitute into the second equation, thereby obtaining a second order equation for x1" obviously won't give you a second order differential equation. I think you have dropped the first 2/3 of the instruction!

    Start by differentiating the [itex]x_1''= -2x_1+ x_2'[/itex]. Now use the second equation to replace that [itex]x_2'[/itex]: [itex]x_1''= -2x_1+ (x_1- 2x_2)= -x_1+ 2x_2.

    NOW solve the first equation for [itex]x_2[/itex] and substitute that into the equation [itex]x_1''= -x_1+ 2x_2[/itex].
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook