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Mathematics Assignment Help

1. Consider the system of ODEs Y 0(t) = AY (t) where Y : [0;1) ! R2 and A =

4 2
2 1

.
(a) Find the eigenvalues, the (proto-)eigenvectors and the eigenvector solutions.
(b) Find the general solution.
(c) What is the equation for the solution curve with initial data (1; 0)?
(d) Sketch the phase portrait including the solution curve with initial data (1; 0).
2. Consider the two-parameter family of linear systems
Y 0(t) =

a 1
b 1

Y (t)
where Y : [0;1) ! R2 and a; b 2 R. In the ab-plane, identify all regions where the system
has a saddle, a sink, a spiral sink, a spiral source, a centre, and in nitely many equilibria.
3. Consider the one-parameter family of linear systems
Y 0(t) =

0 3a
1 a

Y (t)
where Y : [0;1) ! R2 and a 2 R.
(a) Draw the curve in the trace-determinant plane that is obtained from varying the param-
eter a.
(b) Identify the critical values of a in which the phase portrait changes type.

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