In this chapter we introduce first-order *ordinary differential equations* (ODEs) and their numerical approximation in C++. In general, it is difficult to find an analytical solution to these problems and for this reason we use the finite difference method to compute an approximate solution. There is a vast literature on this area of numerical analysis. We scope this domain and we attempt to do justice to the following topics:

- A1: What is an ODE, does it have a solution in a given interval and is the solution unique?
- A2: Qualitative properties of the solution of an ODE.
- A3: Linear and nonlinear scalar ODEs and systems of ODEs.
- A4: Numerical solution of ODEs.
- A5: An introduction to the Boost C++
*odeint*library for solving ODEs. - A6: Some applications of ODEs in computational finance.

We discuss a number of these topics in this chapter. We continue with more advanced cases and applications in Chapter 25. Some relevant literature is Henrici (1962), Brauer and Nohel (1969), Conte and de Boor (1981) and Lambert (1991), to mention just a few. In general, each reader will have her own personal preferences as to which books are most relevant.

An ODE is an equation that defines a relationship involving the derivatives of an unknown function *y* with respect to a single variable. The highest-order derivative in the equation determines the *order* of the equation. ...

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