Partial Solutions to Brainfood #12

1. lambda = -10 +/- sqrt(80). The eigenvectors are (1, lambda).
2. k1 exp(lambda1 t) (eigenvector1) + k2 exp(lambda2 t) (eigenvector2)
3. When t=0, the trajectory is at k1 (1, lambda1) + k2 (1, lambda2). Set theis equal to (1,0) to find k1=1/2 + sqrt(5)/4, k2=1/2-sqrt(5)/4.
4. set x(t)=0.2 and numerically solve for t. I find t=1.5788.
5. In this case k1=1/2+sqrt(5)/2 and k2=1/2-sqrt(5)/2 and the time to get back to x=0.2 is about t=0.75
6. The eigenvalues in this case are complex. You can use numerical simulation, read ahead in the book, use your physical intuition, etc. The shock absorber in this case experiences damped oscillations.