# Linear system of ODEs

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##### New member

Consider the following linear system of ODE :

X’ = -x – y
Y’ = x + 3y
Z’ = 4x + 6y - z

Note that the matrix of this system is exactly the same as
A = [ 1 -1 0
1 3 0
4 6 -1 ]

(a) Study the stability of the fixed point (0,0): is it a source (all solutions diverge to ∞ from it), sink (all solutions converge to it), saddle (any solution is either convergent to the fixed point or diverges to ∞), or neither?

(b) Determine stable and unstable subspaces of (0; 0). (The final answer should be: the stable subspace is spanned by vectors ...., or the stable subspace does not exist.)

(c) Draw a phase portrait of your system in the unstable subspace.

(d) Briefly describe the behaviour of solutions to this system. (e.g. "all the solutions except those in xy-plane will go to ∞ while rotating around z-axis; the solutions that start in xy-plane will stay in that plane and will rotate on the circle centered at the fixed point (0,0) - 5pts. bonus if you can give me a simple matrix of such a system!)

(e) Write down the general solution of the system above using the initial data

x(0) = x0; y(0) = y0; z(0) = z0:

#### MarkFL

Staff member

In part d) of this question, it is stated:

"5pts. bonus if you can give me a simple matrix of such a system!"

This indicates that this question is part of an assignment that contributes to your final grade, and MHB policy is to not knowingly help with graded assignments, which your professor certainly expects to be your work.

Academic honesty is taken very seriously here, and we do not wish to go against the policies of educators by giving help with problems which are meant as assessments of the abilities of their students.

I am directed by policy to close this topic until you can provide me with the contact information for your professor and he/she states in the following correspondence that it is okay for you to seek outside help with this problem.

Best Regards,

Mark.

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