A numerical scheme has been developed to solve the quasilinear form of the transonic stream function equation. The method is applied to compute steady two-dimensional axisymmetric solar wind-type problems. A single, perfect, non-dissipative, homentropic and polytropic gas-dynamics is assumed. The four equations governing mass and momentum conservation are reduced to a single nonlinear second order partial differential equation for the stream function. Bernoulli's equation is used to obtain a nonlinear algebraic relation for the density in terms of stream function derivatives. The vorticity includes the effects of azimuthal rotation and Bernoulli's function and is determined from quantities specified on boundaries. The approach is efficient. The number of equations and independent variables has been reduced and a rapid relaxation technique developed for the transonic full potential equation is used. Second order accurate central differences are used in elliptic regions. In hyperbolic regions a dissipation term motivated by the rotated differencing scheme of Jameson is added for stability. A successive-line-overrelaxation technique also introduced by Jameson is used to solve the equations. The nonlinear equationfor the density is a double valued function of the stream function derivatives. The velocities are extrapolated from upwind points to determine the proper branch and Newton's method is used to iteratively compute the density. This allows accurate solutions with few grid points. The applications first illustrate solutins to solar wind models. The equations predict that the effects of vorticity must be confined near the surface and far away the streamlines must resemble the spherically symmetric solution. Irrotational and rotational flows show this behavior. The streamlines bend toward the rotation axis for rapidly rotating models because the coriolis force is much larger than the centrifugal force. Models of galactic winds are computed by considering the flow exterior to a surface which surrounds a uniform density oblate spheroid. Irrotational results with uniform outward mass flux show streamlines bent toward the equator and nearly spherical sonic surfaces. Rotating models for which Bernoulli's function is not constant show the sonic surface is deformed consistent with the one-dimensional theory.