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Differential equations

  1. Apr 29, 2007 #1
    1. The problem statement, all variables and given/known data

    How would I solve a two-level system such as this:

    [tex]y_1'=-ay_1+by_2[/tex]

    [tex]y_2'=-ay_2-by_1+c[/tex]

    where a b and c are constants.

    2. Relevant equations

    Laplace transform

    3. The attempt at a solution

    I guess my question is what method would I use to solve this? As in should I solve for y1 in terms of y2 and then plug that back into the second equation? Not too sure about the approach.
     
  2. jcsd
  3. Apr 29, 2007 #2
    I would do exactly as you suggested - solve for y2 in terms of y1 and sub back in. You will get a 2nd order ODE, but it is not too difficult. Moreover, if your unsure about the answer, you can always plug your solution back into the equation to ensure your solution satisfies the DE (although this doesn't show you have found all possible solutions). That's one of the great things about them - it's pretty easy to check if your solution is correct.
     
  4. Apr 29, 2007 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    I don't see any way to "solve for y1 in terms of y2" from the first equation because of the [itex]y_1'[/itex] term, nor do I see any reason to use the Laplace transform for such a simple problem. There are, however, several methods you could use. One is to convert from two first order equations to one second order equation: Differentiate the first equation to get [itex]y_1"= -ay_1'+ by_2'[/itex]. Now replace that [itex]y_2'[/itex] with it's expression in the second equation: [itex]y_1"= -ay_1'+ b(-cy_1+ dy_2)= -ay_1'- bcy_1+ d(by_2)[/itex]. From the first equation again, [itex]by_2= y_1'+ ay_1[/itex] so the equation becomes [itex]y_1"= -ay_1'-bcy_1+ dy_1'+ ady_1[/itex] or y_1"+(a-d)y_1'+ (ad- bc)y_1= 0[/itex]. Solve that equation for [itex]y_1[/itex] and then use [itex]by_2= y_1'+ ay_1[/itex] to solve for [itex]y_2[/itex].

    Another way is to write it as a matrix equation:
    If you let
    [tex]Y= \left[\begin{array}{c}y_1 \ y2 \end{array}\right][/tex]
    then your equation becomes
    [tex]Y'= \left[\begin{array}{cc}a & b \ c & d\end{array}\right]Y[/tex]
    You can solve that by finding the eigenvalues and corresponding eigenvectors of the matrix.
     
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