Stability
Example
The solutiony=0 of the system
ay^00 +by^0 +cy= 0
is asymptotically stable if the real part of the characteristic roots is
negative. If the real part of the characteristic roots are non positive and
characteristic roots are unique then the system is stable.
The only case worth considering is whenλ=0 is a double root. That is
whenλ^2 =0, that isy^00 = 0 .In this case the general solution is
y(t)=c 1 +tc 2 which is not stable. The general solution
x(t)=c 1 cosθt+c 2 sinθt
is stable but not asymptotically stable.
