This paper tackles, in essence, two problems. In the first a method of analysis by freezing in intervals of slowly time-varying systems is developed. Under this method the parameters of a time-varying system are frozen in consecutive intervals to their value at the beginning of the interval. Thus in each interval the system becomes time-invariant, or a constant coefficient one, and analytic methods can be applied. Such a mode of solution enables the use of steps which are considerably larger than those common in classical numerical methods. Error estimates are given for the various cases. In the second part the freezing in intervals method is applied to systems with a time-averaged non-linearity. A non-linear system with time-averaging can be regarded as one composed of a time-varying system with a variable parameter. A nonlinear operation, such as squaring, is performed on the signal of the system and followed by low-pass filtering. This filtered signal controls, in turn, the variable parameter. Detailed examples with error analyses are given. Digital simulation by the above and a slower, commonly used, method are included.