Question

Studying for Diff Eq final, can't remember how to do a particular problem...

Answer 1

S'pose the linear system \(\large Y' = AY\) has an eigenvalue -1 with eigenvector \(\large v_1\) and eigenvalue -2 with eigenvector \(\large v_2\) where the 2 vectors are shown below.



A) Sketch the phase plane, including the solution through (0,1)
B) Sketch the x(t) and y(t) graphs of the solution passing through (1,0) at t = 0

Answer 2

Oh wait, I think I'm making this out to be harder than it is

These are the vectors...and I am given my eigenvalues so my general solution would be

\[\large k_1e^{-t}v_1 + k_2e^{-2t}v_2\]

Then we just see what happens as t goes to infinity, which dominates etc...okay, figured it out lol

Answer 3

So in this case, both are pointing in...and the v2 dominates so my phase portrait would look like...



or something along those lines I believe