The author considers initial value problems connected with nonlinear differential equations of the second degree, say , with initial data of the form , where and stand for given real constants while is the Heaviside unit step function, and gives a method to find an approximate weak solution valid in the asymptotic sense when certain parameter tends to zero. This approximation requires to replace by a function such that for any . The method is based on the identity
where stands for any function such that its derivative belongs to the Schwartz space and , , while refer to two functions defined in terms of and . As to , it denotes any distribution, say , such that . The method is applied to the classical Hopf equation for which . An extension of the method to the case when belongs to a two-dimensional space is also shown on a particular example. Remark that this second application contains some misprints.