8 Ordinary differential equations III

1. New: 3.4.14 Old: 3.4.14

2. New: 3.5.2 Old: 3.5.2 Graph neatly.

3. New: 3.5.8 Old: 3.5.8

4. For the system of New: 3.6.16 (a) Old: 3.6.16 (a), solve both the given inhomogeneous problem with zero initial conditions and the homogeneous problem for arbitrary initial conditions. ``Solve'' means here find $\hat y_1$ and $\hat y_2$. Do not try to find $y_1$ and $y_2$ themselves. Assume the constant $c_1$ is a viscous damping constant, even though it looks like dry friction.

5. Now address part (b) of the same question. Hint 1: look at the form of the partial fraction solution to see the qualitative response of mass $M$. Hint 2: some of the roots of the denominator may be hard to identify mathematically. Use the physics instead. Based on energy considerations, what can you say about the long-term behavior of the homogeneous equations? So what does that say about the roots of the denominator for the homogeneous solution?