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Homework Help: 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
    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].
     
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