Differential Equations Explained
Model change with differential equations, visualize phase portraits, and compare numerical methods like Euler and Runge–Kutta.
📉 Phase Portrait
📚 Fundamentals
- Autonomous systems: x' = f(x)
- Equilibria: f(x*) = 0; classify by linearization
- Stability: eigenvalues of Jacobian
Numerical Integration
Euler is simple but unstable for stiff systems; RK4 is a robust default for smooth dynamics.
❓ Frequently Asked Questions
1) What is stiffness?
Widely separated time scales cause instability for explicit methods.
Widely separated time scales cause instability for explicit methods.
2) When to use implicit methods?
Stiff problems benefit from A-stable schemes like backward Euler.
Stiff problems benefit from A-stable schemes like backward Euler.
3) Can ODEs be chaotic?
Yes; see Lorenz equations.
Yes; see Lorenz equations.
4) Phase vs time plots?
Phase shows geometry; time plots evolution of components.
Phase shows geometry; time plots evolution of components.
5) Accuracy control?
Adaptive step sizes maintain error tolerances.
Adaptive step sizes maintain error tolerances.
6) Systems vs scalar ODEs?
Vector fields vs single dimension; same principles.
Vector fields vs single dimension; same principles.
7) Existence/uniqueness?
Lipschitz conditions guarantee well-posedness locally.
Lipschitz conditions guarantee well-posedness locally.
8) PDEs vs ODEs?
PDEs depend on space and time; ODEs on time only.
PDEs depend on space and time; ODEs on time only.
9) Linear vs nonlinear?
Linear superposition holds only for linear systems.
Linear superposition holds only for linear systems.
10) Tools?
SciPy, DifferentialEquations.jl, MATLAB ODE suite.
SciPy, DifferentialEquations.jl, MATLAB ODE suite.