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

Out[13]=

                           2
  {P'[t] == a P[t] - s P[t] }

Out[14]=

  P[t]

Out[15]=

  t

In[16]:=

  messysolution=DSolve[equation,depvar,indepvar]

Out[16]=

                   a t
                a E
  {{P[t] -> ---------------}, {P[t] -> 0}}
             a t
            E    s + a C[1]

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