e is an asymptotically stable proper node f is an unstable i

(e) is an asymptotically stable proper node. (f) is an unstable improper node. I can\'t tell what makes one stable and the other unstable. Please explain clearly. Thanks!

(e) The general solution is x(t) = Ce-5t | C2e_5t (f) The general solution is x(t) = Get-1 | + Cget I-

Solution

(e)

Here the solution has two independent components. Each component is multiplied by: e^{-5t}

When we talk of asymptotic stability we are talking of large time behaviour. And e^{-5t} at large time goes to 0 hence asymptotically stable solution

f)

In this case though both components are multiplied by e^t which goes to infinity for large time and hence unstable solution.