Ordinary Differential Equations
Ordinary differential equations (ODEs) describe systems evolving in time. They appear throughout physics, biology, control theory, and machine learning dynamics.
Key Points
- Existence and uniqueness are guaranteed under Lipschitz continuity (Picard-Lindelöf).
- Linear ODEs can be solved via eigenvalue/eigenvector methods.
- Numerical methods (Euler, Runge-Kutta) approximate solutions.
Formulas
First-order linear ODE
$$\frac{dy}{dt} = a(t)y + b(t)$$
Picard-Lindelöf
$$\frac{dy}{dt} = f(t,y), \quad y(t_0)=y_0 \text{ has unique solution if } f ext{ is Lipschitz in } y$$
Euler method
$$y_{n+1} = y_n + h f(t_n, y_n)$$
Code Example
from scipy.integrate import solve_ivp
def dy_dt(t, y):
return -0.5 * y
sol = solve_ivp(dy_dt, [0, 10], [2.0], t_eval=np.linspace(0, 10, 100))
print(sol.y[0, -1]) # decays to ~0.09