A new method for solving the compressible, laminar, boundary layer equations over a flat plate is developed in the present study.. The method accounts for the variations of all boundary . layer parameters which control the flow, and it is able to
solve the equations for any gas. In this method, the Illingworth transformation is used to transform the equations of motion to a set of ordinary differential equations (ODE). The fourth-order Runge-Kutta method in conjunction with the shooting method are used to solve the set of ODE. A very efficient procedure is developed to find out the unknown boundary conditions at the wall. The method is applied to investigate the characteristics of supersonic flow of air over cooled, adiabatic, and heated walls, where wide ranges of free stream Mach numbers and wall-free stream temperature ratios are used. Comparsons of the present results with previous ones show that the method is very efficient. In addition to the verification of the success of the method, investigation of the characteristics of the flow leads to very useful results.