Complex eigenvalues - solve the system

[schmiggy]

Homework Statement

Using eigenvalues and eigenvectors, find the general solution to
dx/dt = x - y
dy/dt = x + y

Homework Equations

Matrix 'A' - lambda*identity matrix ; for finding eigenvalues and thus eigenvectors

Other relevant equations written on the attached scanned image of my attempt at solving the question.

The Attempt at a Solution

Attached is my attempt, my lecture notes aren't clear on which eigenvalue to use when determining a general solution so at first I used the eigenvalue lambda = i + 1 which yielded a solution far from that in the answer section of this work booklet.

Using lambda = -i + 1 I got an answer very similar to the correct answer. At the bottom surrounded by a scribbled box is the answer from the book however I'm confused how they got their imaginary values...

 

[ehild]

Note that 1-(i+1)=-i instead of 2-i.

ehild

 

[schmiggy]

Note that 1-(i+1)=-i instead of 2-i.

ehild

Ugghhh how embarrassing.. sorry, stupid error. I'll fix that up in the morning.. how about the second lot of calculations though? I'm pretty sure I got the basic arithmetic right on them.. still the answer is wrong.
Thanks

 

[schmiggy]

Ok, I've fixed my atrocious basic level maths from last night and got the correct answer.. I'm honestly not sure how I made such a terrible mistake but I did.. sorry if I wasted anyone's time.

My question remains however, is there only one general solution to a problem? For example using the eigenvalue i + 1 gives me the general solution as per the back of the book, however using the eigenvalue -i + 1 gives me an answer close to the one at the back but not quite.

When calculating a general solution are we supposed to only use the eigenvalue where 'i' is positive??