# Find State Transition Matrix (time variant)

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1. Oct 5, 2014

### johnpjust

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. Oct 6, 2014

### Ray Vickson

The two columns $\vec{x}_1$ and $\vec{x}_2$ of $X$ are solutions of
$$\frac{d}{dt} \pmatrix{x\\y} = \pmatrix{-1 &\exp(-t - t^2/2)\\0 & t } \pmatrix{x\\y}$$
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