Find State Transition Matrix (time variant)

[johnpjust]

Homework Statement

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)

Homework Equations

How to go about solving such problems in a systematic way?

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!

 

[Ray Vickson]

Homework Statement

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)

Homework Equations

How to go about solving such problems in a systematic way?

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!

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.