Yes, p is a function of t, and a is a fixed constant (even though its value could be any arbitrary positive number)
differential operator is LINEAR so one can always pull a constant factor out from a certain function, namely
d(cf) = c df where f is a function and c is a constant.
I strongly recommend you review some basic facts in calculus.
Integration is done by noticing that the derivative of ln(ln(x)) is 1/(x lnx), an exercise for first-year calculus students.
Domain is defined at the very beginning of the problem: k and a are positive values.
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