Mathematical Methods and Mechanics

(a)

Question 5

The question concefl$ the system of differelrtial equdions

(b)

(c)

{ *=ty-xy . I j'= 2x-ry

Write the system inthe form

l*,t|r = Y(x,y)

where V(x, y) is a vector field. (2 marks)

Find all the equilibrium points for fte system of differential equations. (6 mmks)

For each of the equilibrium points (Xu Yil define the exsesses u and v byr = u* Xo and y=v + Ioand usettreseto find alinear approximation t = M u for the system of equations near the equilibrium point.

(7 marks)

ln each case find the eigenvalues for the mafix M. Hence classifu each equilibrium point and sketch the vector fields in the neighbotrhood of that point.

(10 marks)

(d)