We have computed an extensive grid of binary evolution tracks to represent low- and intermediate mass X-ray binaries ( LMXBs and IMXBs ) . The grid includes 42,000 models which covers 60 initial donor masses over the range of 1 - 4 M _ { \odot } and , for each of these , 700 initial orbital periods over the range of 10 - 250 hours . These results can be applied to understanding LMXBs and IMXBs : those that evolve analogously to CVs ; that form ultracompact binaries with P _ { orb } in the range of 6 - 50 minutes ; and that lead to wide orbits with giant donors . We also investigate the relic binary recycled radio pulsars into which these systems evolve . To evolve the donor stars in this study , we utilized a newly developed stellar evolution code called “ MESA ” that was designed , among other things , to be able to handle very low-mass and degenerate donors . This first application of the results is aimed at an understanding of the newly discovered pulsar PSR J1614-2230 which has a 1.97 M _ { \odot } neutron star , P _ { orb } = 8.7 days , and a companion star of 0.5 M _ { \odot } . We show that ( i ) this system is a cousin to the LMXB Cyg X-2 ; ( ii ) for neutron stars of canonical birth mass 1.4 M _ { \odot } , the initial donor stars which produce the closest relatives to PSR J1614-2230 have a mass between 3.4 - 3.8 M _ { \odot } ; ( iii ) neutron stars as massive as 1.97 M _ { \odot } are not easy to produce in spite of the initially high mass of the donor star , unless they were already born as relatively massive neutron stars ; ( iv ) to successfully produce a system like PSR J1614-2230 requires a minimum initial neutron star mass of at least 1.6 \pm 0.1 M _ { \odot } , as well as initial donor masses and P _ { orb } of \sim 4.25 \pm 0.10 M _ { \odot } and \sim 49 \pm 2 hrs , respectively ; and ( v ) the current companion star is largely composed of CO , but should have a surface H abundance of \sim 10 - 15 \% .