Important correction from the lecture!

At the last part of this Tuesday's lecture I had a wrong interpretation of what the word "stability" means.

I said that we have stability when both the general and particular solution converges to some number (and in this case the general solution would go to zero), but this is not correct!

To say that a solution is stable, all we need is the homogeneous version to go to zero, the behaviour of the particular solution is irrelevant.

The reason for this is that stability in this context is with respect to whether or not the solution is strongly affected by changes in the initial values x_0 and x_1. If we know these values, then A and B will be specific constants depending on these, and the effect of an change in x_0 and x_1 is equivalent of asking the effect of an change in A and B. And such changes will not have an explosive effect if and only if u1 and u2 converges, which in this context only happens if they go to zero. Thus, the system is stable if and only if the general solution goes to zero. Therefore, you can ignore the small extra discussion I had about the behaviour of the particular solution, for stability it is ok that this goes to infinity, since we simply are concerned about the effect of changes in the initial values x_0 and x_1, and these do not have an effect on the particular solution.