Integro-differential equations are derived which govern the spatial distribution of ions of energyE' moving in a solid target after the ions were incident on the target surface with energyE. These equations are specialized to integral equations which govern the first and second spatial moments of the distribution. Solutions to the moments equations are obtained to second order for an expansion of the integrands in powers ofT, the energy transfer in a single collision between an ion and an atom (or electron) of the target. Similar equations are also derived for the average range along path of the ions at energyE'.