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#### ZiniaDuttaGupta

##### New member

- Jun 12, 2013

- 3

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: