# 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 innitely 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.