General Solution in terms of the parameters a and s.

  Clear[equation,a,s,messysolution];
  equation={P'[t]==a P[t] - s (P[t])^2}
  depvar=P[t]
  indepvar=t

  messysolution=DSolve[equation,depvar,indepvar]

Note that Mathematica gives two solutions. The second one is a trivial solution, so the equation is homogeneous. The first one can be obtained by separation of variables.

Exercise 1

Exercise 2