The characteristic impedance of a transmission line, a wire or a conductor changes in a non-uniform manner if the distance to the ground at all points longitudinally is not the same. Vertical conductors, transmission towers and sagging overhead lines are examples for the non-uniform lines. In this study, lumped-parameter-based state variable representation of the single-phase non-uniform line is described. From the lumped-parameter non-uniform line model a linear set of first-order differential equations is obtained in the form of state equations and this analytical expression is solved in closed form using MATLAB to obtain the transient response directly in the time domain. The closed-form solution has the advantage of obtaining the response of the system at an instant without the need for data in the previous states except for the initial conditions. The method also allows attaining the voltage and current profile of the system for any instant. In the illustrative cases presented, the systems with different surge impedance variations are considered and the surge response of a vertical conductor, an exponential line, and a horizontal cone and a vertical cone with constant and also varying propagation velocity are computed. The results are verified by those obtained using s-domain simulations of distributed-parameter transmission line and inverse Laplace transform.