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Solving Differential Equations in Python

Sep 21, 2025

Overview

This lecture introduces solving differential equations in Python using SciPy, covering first-order, coupled, and higher-order ODEs, and emphasizes proper equation structuring and solver selection.

First-Order Differential Equations in Python

  • Most ODE solving in Python is done with SciPy, using functions odeint and solve_ivp.
  • Equations must be written in the form: derivative = function of time and the variable, e.g., dv/dt = f(t, v).
  • Define the ODE as a Python function taking time and variable as arguments.
  • Initial conditions are specified (e.g., v(0) = 0).
  • Both odeint and solve_ivp require the function, initial condition, and time span/array.
  • odeint expects the function (with t first if tfirst=True), initial value, and times as arguments.
  • solve_ivp is more customizable, requiring the function, a time span, initial value (as an array), and evaluation points.

Working with Solution Objects

  • Both solvers return objects containing arrays of solutions, structuring data for multi-variable systems.
  • Extract single-variable solutions by indexing or transposing the result arrays.

Coupled First-Order Differential Equations

  • Systems are written with each variable having its own ODE, often with mutual dependencies.
  • Package all variables into a vector (e.g., S = [y1, y2]).
  • Define a function returning a vector of derivatives, given the current vector and independent variable.
  • The pattern is easily extended to larger systems by expanding the size of the vector.

Higher-Order Differential Equations

  • Any higher-order ODE must be rewritten as a system of first-order ODEs.
  • For a second-order ODE, introduce new variables to represent each derivative up to one fewer than the highest order.
  • Example: x'' = f(x', x, t) becomes two first-order ODEs using v = x'.
  • For nth-order (or coupled systems), define new variables recursively for each derivative.

Setting Up and Solving Large Systems

  • Package all variables into a state vector S (e.g., for 3rd order: [x1, v1, a1, x2, v2, a2]).
  • Define a function returning derivatives for each component, extracting and updating based on definitions.
  • Specify all initial conditions in a vector and solve using the same routines.

Precautions for Complex or Chaotic ODEs

  • Not all solvers work for every system; some ODEs, especially chaotic ones, require specific solvers and parameter tuning.
  • For sensitive systems, use methods like 'DOP853' in solve_ivp and set tight tolerances (atol, rtol).
  • Review the relevant literature for recommended solvers and settings.

Key Terms & Definitions

  • ODE (Ordinary Differential Equation) — An equation involving derivatives of a function with respect to one variable.
  • Initial Condition — The value of the function (and possibly its derivatives) at the starting point.
  • odeint — SciPy function for solving ODEs, uses ODEPACK's LSODA solver.
  • solve_ivp — SciPy function for solving initial value problems for ODEs, supports multiple solvers.
  • Coupled ODEs — A set of differential equations where each equation depends on multiple unknown functions.
  • Tolerance (atol, rtol) — Parameters controlling numerical precision during ODE solving.

Action Items / Next Steps

  • Practice converting higher-order ODEs to systems of first-order ODEs.
  • Experiment with both odeint and solve_ivp for different ODE systems.
  • Research solver requirements and parameter tuning for any complex or chaotic system you need to solve.

Full transcript