1. The problem statement, all variables and given/known data (Everything here is frequency dependant, but I have left off the v from all the variables to minimise chance of typos - hopefully that is ok but please let me know if not and I will try and retype!) The equation of radiative transfer is dI/dτ = -I + S We guess that I(τ) = f(τ)ebτ Taking the derivative of this gives dI/dτ = fbebτ + ebτdf/dτ which is just the same as dI/dτ = bI + ebτdf/dτ Substituing this back into the first equation gives bI + ebτdf/dτ = -I + S So b = -1 and S = e-τdf/dτ I have been able to follow all this but the next step is where I get lost - my lecture notes go straight to f = ∫Setdt + c0 (Integral limits from 0 to τ) "where t is a dummy variable". 2. Relevant equations All given above I think 3. The attempt at a solution My approach was to go from S = e-τdf/dτ df/dτ = Seτ Separate variables df = Seτdτ Then integrate from 0 to f and from 0 to τ f = ∫Seτdτ But I'm not sure how to do that integral and (according to my notes) it is the wrong approach anyway! If someone could fill in the gaps and take me step-by-step through the method used in the notes it would be really helpful. I don't get how this other variable t is allowable or useful? Or where the c0 comes from?