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Homework Help: Find State Transition Matrix (time variant)

  1. Oct 5, 2014 #1
    1. The problem statement, all variables and given/known data
    find the state transition matrix of a time varying system where:
    dX/dt = A*X

    with A = [-1 , exp(-t - (t^2)/2) ; ; 0 , t] (Matlab format - sorry but its easier)

    2. Relevant equations
    How to go about solving such problems in a systematic way???

    3. The attempt at a solution
    I've found eigen values of (-1) and (t), and associated eigen vectors of [1 ,0]T and [exp(-t - (t^2)/2) , (t+1)]T......which lead me to one solution vector of [exp(-x) , 0]T....and having trouble getting the other one. HOWEVER, i'm just "bushwhacking" here....I need some help in figuring out a systematic approach.

    Thanks!!
     
  2. jcsd
  3. Oct 6, 2014 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    The two columns ##\vec{x}_1## and ##\vec{x}_2## of ##X## are solutions of
    [tex] \frac{d}{dt} \pmatrix{x\\y} = \pmatrix{-1 &\exp(-t - t^2/2)\\0 & t } \pmatrix{x\\y} [/tex]
    Expanding it out, the DE for ##y## is simple enough to solve; then substitute the solution ##y(t)## into the first equation and solve that.
     
    Last edited: Oct 6, 2014
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