# Chapter 10

# 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 ...