Find the equations for the trajectories for case wb a gt 00

Question

1. Reduce the following matrices to Jordan form.

2. Given

where 4 is a Null (all Os) Vector.

(a) Show that the above equation is of "classical normal modes" type, i.e. it can be uncoupled.

(b) Solve using 2-N space analysis.

3. Determine the component of exp (A t) where A = 33 is a Jordan form of order 3.

4. Show that there exist unbounded solutions to the system of equations

(Hint: Consider Substitution x=tz)

5. Find the trajectories in the phase plane of the differential equation:
u^.. +u = {k/(1-u)} k > 0 is a constant. Include the equations for the trajectories, and, in 1-u particular, that of separatrix.

6. Find the equations for the trajectories for Case WB (a > 0.0 0. a2 > 40) and Case IVC (α >0. β> 0. α² = 4β). Are the straight lines indicated in class trajectories? What are the equations describing them?

7. The differential equation describing the motion of a simple pendulum is given by:
u^.. + ω0²sinu =0
Derive an expression for the period of vibration in terms of elliptic integrals. Find the series expansion of T correct to second power in amplitude.

8. Given the equation

i. For what set of initial conditions are solutions to the above equation periodic?

ii. If b = 0. use the Lindsted-Poincare perturbation method to obtain the periodic solution of the above equation correct to second order in a.

iii. Is the time average of the above solution zero? Explain.