This paper presents solutions for the classical one-dimensional (1D radial and Cartesian) problem of Langmuir probes in a collisionless, isothermal plasma. The method is based on two-fluid equations derived from the first two moments of Vlasov's equation. In contrast to commonly used approximations, electron inertia and ion temperature are not neglected so that the fluid equations are symmetric in terms of electrons and ions. The fluid equations are reduced analytically so that the electric potential is the only remaining spatial function, which is numerically determined using Poisson's equation. The single radial solution applies continuously over the whole region from the probe up to the unperturbed plasma, in contrast to theories which separate the probe boundary region into a charged sheath and a quasi-neutral pre-sheath, and is valid for all values of probe bias potential. Current-voltage characteristics are computed for cylindrical and spherical probes, which exhibit non-saturation of the ion and electron currents. The 1D Cartesian case is also analysed, and the Bohm criterion is recovered only in the limit of large radius probes. Published by AIP Publishing.