We study the Cauchy problem for an almost linear first order differential equation with discontinuous coefficient of the time-derivative. We analyze differential properties of a weak solution to the forward problem and study the inverse problem of finding the discontinuity lines for the time-independent coefficient. For this purpose the trace of a solution to the Cauchy problem is assumed to be given on a plane far from the line in question. We develop an algorithm for solving the inverse problem. Our study can be considered as a fragment of the theory of probing inhomogeneous media by physical signals.