Page - 203 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

8SolvingOrdinaryDifferentialEquations Differential equations constitute one of the most powerful mathematical tools to understand and predict the behavior of dynamical systems in nature, engineering, andsociety.Adynamicalsystemissomesystemwithsomestate,usuallyexpressed by a set of variables, that evolves in time. For example, an oscillating pendulum, thespreadingofadisease, and theweatherareexamplesofdynamicalsystems. We can use basic laws of physics, or plain intuition, to express mathematical rules that govern the evolution of a system in time. These rules take the form of differential equations. You are probably well experienced with equations, at least equations likeax+ b = 0 and ax2 + bx+ c = 0, where a, b and c are constants. Such equations are known as algebraic equations, and the unknownsare numbers. In a differential equation, theunknownisa function, andadifferentialequationwillusually involve this function and one or more of its derivatives. When this function depends on a single independentvariable, the equation is calledanordinarydifferential equation (ODE, plural: ODEs), which is different from a partial differential equation (PDE, plural: PDEs), in which the function depends on several independent variables (Chap. 9). As an example, f ′(x) = f(x) is a simple ODE (asking if there is any function f such that it equals its derivative—you might remember that ex ©The Author(s) 2020 S.Linge, H.P. Langtangen, Programming forComputations -Python, Texts in Computational Science and Engineering 15, https://doi.org/10.1007/978-3-030-16877-3_8 203