In this paper the approximation which was originally introduced by Chaplygin in his famous paper "on gas jet" for the treatment of subsonic flow is extended to the supersonic motion. Four typical initial value problems are solved using hodograph variables, and the analytical forms of the solutions are obtained. The approximate solutions of supersonic flow given by Christianovich is extended in this paper to slightly supersonic flow. Finally the relation between the Christianovich's approximate method and the general treatment presented here is studied.