three counter rotating wheels will slip against neighboring wheels thus experiencing frictio

The three, counter-rotating wheels will

slip against neighboring wheel(s), thus experiencing friction, if the angular

speeds are not commensurate with the wheels rolling together. The rotation

of the wheels is governed by the following system of ODE’s:

x’1 = -x1 – 2×2

x’2 = -3×1 – 12×2 – 2×3

x’3 = -3×2 – x3 ;

where xj is the frequency of the j-th wheel (in cycles per second, with

clockwise positive and counter-clockwise negative) and the over-dot indicates

the derivative with respect to time.

(a) Find the solution of the system subject to the initial values

x1(0) = 5 ; x2(0) = 0 ; x3(0) = 0 ;

and use a plotting device to illustrate x1(t), x2(t), and x3(t) together.

(b) Find the solution of the system subject to the initial values

x1(0) = 0 ; x2(0) = 0 ; x3(0) = 5 :

and use a plotting device to illustrate x1(t), x2(t), and x3(t) together.

system (for different initial conditions) as t goes to +infinity.