Question

x’(t) = x(t) – αy(t)
y’(t) = x(t) + y(t)

For which values of α do there exist straight trajectories and find the equation of the lines which contains them.

Answer 1

I believe this is eigenvalue problem

y(t)=x'(t)/a-x(t)/a

y'(t)=x''(t)/a -x'(t)/a

plug that into second equation

x''(t)/a -x'(t)/a= x(t) +x'(t)/a-x(t)/a

Answer 2

simplify it then solve the differential equation

Answer 3

dedinitely an eigenvalue problem, but I'm used to doing it with matrices. I get an eigenvalue containing α and can't get an eigenvector unless I give α a value.

Answer 4

what do you mean by linear trajectory? do you mean you are seeking the solution of type
y = mx

Answer 5

yeah!

Answer 6

looking at this applet, my best guess is it depends on initial condition
http://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-iv-first-order-systems/qualitative-behavior-phase-portraits/LinPhasePorCursor.html
and you should have real eigen vectors

Answer 7

The real eigenvalues are possible for α less than three. which leaves me with quite a few options. I'm pretty sure I get two independant eigenvectors for each eigenvalue. If you plug in -1 for α you get one E-vector for each one. I'm completely lost. hahaha

Answer 8

looks like more can be found here
http://tutorial.math.lamar.edu/Classes/DE/PhasePlane.aspx