What's Next?

Congratulations

You have completed all 15 lessons. You can now solve differential equations numerically — both scalar ODEs and systems — and analyze their qualitative behavior through equilibria and stability.

What to Explore Next

  • SciPy -- scipy.integrate.solve_ivp uses adaptive RK45 with error control. It handles stiff equations and automatic step-size selection far beyond what we built here.
  • Stiff equations -- Some ODEs require implicit methods (like backward Euler or the Runge-Kutta implicit family). Stiffness arises in chemical kinetics, electronics, and more.
  • Partial differential equations (PDEs) -- ODEs involve one independent variable (time). PDEs involve space and time: heat equation, wave equation, Navier-Stokes.
  • Chaos theory -- The Lorenz system (dx/dt = σ(y-x), etc.) shows how deterministic ODEs can produce unpredictable, chaotic behavior.
  • Bifurcation theory -- How equilibria appear, disappear, or change stability as parameters vary.

References

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