# Ordinary Differential Equations: Initial-Value Problems

Core Topics

Euler's methods (explicit, implicit, errors) (10.2).

Modified Euler method (10.3).

Midpoint method (10.4).

Runge–Kutta methods (second, third, fourth order) (10.5).

Multistep methods (10.6).

Predictor-corrector methods (10.7).

Systems of first-order ODEs (10.8).

Solving a higher order initial value ODE (10.9).

Use of MATLAB built-in functions for solving initial-value ODEs (10.10).

Complementary Topics

Local truncation error in second-order Runge–Kutta method (10.11).

Step size for desired accuracy (10.12).

Stability (10.13).

Stiff ODEs (10.14).

## 10.1 BACKGROUND

A differential equation is an equation that contains derivatives of an unknown function. The solution of the equation is the function that satisfies the differential equation. A differential equation that has one independent variable is called an ordinary differential equation (ODE). A first-order ODE involves the first derivative of the dependent variable with respect to the independent variable. For example, if x is the independent variable and y is the dependent variable, the equation has combinations of the variables x, y, and . A first-order ODE is linear, if it is a linear function of y and (it can be a nonlinear function of x). Examples ...

Get Numerical Methods for Engineers and Scientists 3rd Edition now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.