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Problem in finding relationship of ordinary differential equation

Hi guys, I am currently working on a set of non-linear ordinary differential equation which is my laser model. Attached is the equations and respective derivation.

I need to find the relationship between N(t) and S(t). At equation (5), if I could find the relationship between this equation to be zero, then I could directly find the N(t) and S(t) relationship. There is relationship between \tau n and \tau p because the theory behind this equations are conservation of energy and definitely there is a relationship between them. Can anyone please advice me on how to solve this final part. Thank you very much!!