Differential Equation. Help, please.
[tex]\frac{dP}{dt}=aP-bP^2=bP(\frac{a}{b}-P)\\\\P' - aP = -bP^2\ \ Bernoulli:\ P' + f(t)P=g(t)P^n\\\\Substitute:\\ u=P^{1-n}=P^{-1}\\du=-P^{-2}dP\\(-y^2)du=dP\\u'=\frac{du}{dt}\\\\(-P^2)u'-aP=-bP^2\\u'-aP(-P^{-2})=-bP^2(-P^{-2})\\u'+aP^{-1}=b\\Substitute: u'+au=b\ \ Linear: u' + f(t)u=g(t)\\I=e^{\int a\ dt} = e^{at}\\e^{at}u'+ae^{at}u=be^{at}\\e^{at}u = \int be^{at}dt\\e^{at}u=\frac{b}{a}e^{at}\\u=\frac{b}{a}\\\frac{1}{P}=\frac{b}{a}\\P=\frac{a}{b}[/tex]
What have I done wrong here?
