From the introduction: The theory of generalized functions is one of the most powerful tools for investigating linear differential equations. However, from the very beginning, the distribution theory has an essential disadvantage: it is inapplicable to nonlinear problems. Therefore, different interpretations of the solution of the nonlinear differential equations were proposed. Unfortunately, different interpretations of the same equation lead, in general, to different solutions.
In this paper, we consider the following nonlinear equation with generalized coefficients \[ \dot X(t)=f(X(t)) \dot L(t)+ g(X(t)), \quad t\in [a,b], \] where \(\dot L(t)\) is a derivative of the function of finite variation in the distributional sense. We use the algebra of new generalized functions from Yu. V. Egorov [Russ. Math. Surv. 45, 1–49 (1990; Zbl 0754.46034); translation from Usp. Mat. Nauk 45, No. 5 (275), 3–40 (1990)]. The main purpose is to show that under some conditions, the solution (as a new generalized function) associates with some ordinary function which is natural to call the solution of this equation. Also it is shown that the solutions of the above equation in the sense of the previous approaches can be obtained from the solution of the equation in the differentials in the algebra of new generalized functions.