Rewriting logistic equation

Question

We are given the logistic equation: $\ \frac{dp}{dt}= [a-b\cdot p(t)]\cdot p(t)\,\,\, t>0 \,\,\,\,\,\,(1) $

And we want to solve it by rewriting it to: $\ \frac{dq}{dt} = q(t) - q(t)^2, \,\,\, t>0 \,\,\,\,\, (2)$.

with $\ q(t) = \frac{b}{a}\cdot p\left(\frac{t}{a}\right) \,\,\,\,\,\,(3)$

The first step or exercise is to show that I can rewrite equation (1) to equation (2) by using (3).

I know that if I multiply (3) by $\frac{a}{b}$ and then plug it into (1) I get to equation (2). However, If I do that, I am actually plugging in for $p\left(\frac{t}{a}\right)$ and not for $p\left(t\right)$.

So how can I rewrite $p\left(\frac{t}{a}\right)$ to just $p\left(t\right)$

Thanks for help and sorry for the probably dumb question :/

Answer 1

You could just differentiate using the chain rule or first transform using $s=t/a$ to $$p(s)=\frac{a}{b}·q(as)$$ so that $$p'(s)=\frac{a^2}{b}·q'(as).$$