SOLVING THE LOGISTIC
DIFFERENTIAL EQUATION

> restart:with(DEtools):

> LogisticDE:=diff(y(t),t)=a*y(t)-b*y(t)^2;

> dsolve(LogisticDE,y(t));

That was easy! Let us try to get it into standard form using various substitutions.

> subs(a=M*b,%);

> simplify(%);

> subs(b=a/M,%);

> subs(_C1=K/M,%);

> LogisticEq:=y=M/(1+K*exp(-a*t));

> subs({y=y0,t=0},%);

> solve(%,K);

> subs(K=(M-y0)/y0,LogisticEq);

The above is standard form.