These are known as the two-stream equations, and will serve as the basis for all subsequent discussion of radiative transfer in this book, save that we will incorporate the neglected scattering term in Chapter 5.
Because of the neglect of scattering, the equations for I+ and I_ are uncoupled, and each consists of a linear, inhomogeneous first order differential equation. The solution can be obtained by substituting I+ = A(tv) exp(-tv), and similarly for I_, which reduces the problem to evaluation of a definite integral for A. The result is
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