ABSTRACT
This chapter provides the exact solution of the ordinary differential equation (ODE). It resolves the ODE with the forward Euler method. Non-linear stiff equations occur throughout applications where we have two (or more) different timescales in the ODE, that is , if different parts of the system change with different speeds. A typical example are equations of chemical kinetics where each timescale is determined by the speed of reaction between two compounds. Such speeds can differ by many orders of magnitude. The basic input of a well-designed computer package for ODEs is not the step size but the error tolerance, as required by the user. The chapter discusses multistep methods which can be expressed as linear combinations of approximations to the function values and first derivatives.
