14.6 Review of the course
The course has followed one thread: from the explicit solution of special equations to the qualitative understanding of general ones.
| Question | Main tools (sections) |
| Does a solution exist, is it unique? | Picard–Lindelöf, Peano, Gronwall (2); maximal solutions, blow-up (3) |
| How does it depend on the data? | |
| What do linear systems do? | Fundamental matrices, Liouville (4); \(e^{At}\), Jordan form (5); Laplace (6) |
| Periodic coefficients? | Floquet multipliers, Hill, Mathieu (7) |
| Variable coefficients? | |
| Is an equilibrium stable? | |
| Are there oscillations? | |
| What happens as parameters vary? | Saddle-node, pitchfork, Hopf; chaos (13) |
| How do we compute it? | Perturbation methods; Runge–Kutta, stiffness (14) |
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