The concept of stability is loosely defined as follows: If there exists an for each differential equation such that a change (perturbation) in the starting value by a fixed amount produces a bounded change in the numerical solution for all then the method is stable. In other words, let be the solution of (2.1) with initial condition and let be the solution of the same method (2.1) with a perturbed initial condition . Then the method (2.1) is stable if there exists positive constants such that

whenever .

This definition will be modified when multistep methods are discussed.

Note: We also see that stability is related to a method and well posedness is related to a problem.

The Euler's method for solving

is given by

(2.16)