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General solution of differential equation system

  1. Oct 29, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the general solution of the system of differential equations
    [tex]x'=10x - 12y[/tex]
    [tex]y'=25x - 30y[/tex]
    (where primes indicate derivatives with respect to t) by using the initial conditions
    [tex]x(0)=A[/tex]
    [tex]y(0)=B[/tex]

    2. Relevant equations

    3. The attempt at a solution

    [tex]x''=10x' - 12y'[/tex]
    [tex]y'=25x-30y[/tex]
    [tex]x''=10x'-12(25x-30y)[/tex]
    [tex]y=\frac{10x-x'}{12}[/tex]
    [tex]x''=10x'-300x+360(\frac{10x-x'}{12})[/tex]
    [tex]x''=10x'-300x+300x-30x'[/tex]
    [tex]x''+20x'=0[/tex]
    [tex]r^2+20r=0[/tex]
    [tex]r(r+20)=0[/tex]
    [tex]r=0,-20[/tex]
    [tex]x(t)=Ae^{-20t} +B[/tex] [tex]\rightarrow[/tex] [tex]x(0)=A[/tex]
    [tex]A=A+B[/tex]
    [tex]B=0[/tex]
    [tex]y=\frac{10x-x'}{12}=\frac{5}{2}Ae^{-20t} \rightarrow [tex]y(0)=B[/tex]
    [tex]0=\frac{5}{2}A[/tex]
    [tex]A=0[/tex]

    If A=0 and B=0, then my general solutions x(t),y(t)=0. This is clearly not right. What am I doing wrong? Thanks!
     
  2. jcsd
  3. Oct 29, 2009 #2

    LCKurtz

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    Use C and D in your general solution so they don't get mixed up with the A and B in the initial conditions.
     
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