# How do I scale the logistic differential equation?

My lecturer kinda brushed over this but my constants just aren't cancelling out properly.

For the 'ordinary' logistic differential equation we have $dN/dt= N(t)[a - bN(t)]$. He scales this by replacing $N$ with $bN/a$ and $t$ with $at$. In the end he ends up with $dN/dt = N(t)[1 - N(t)]$. I tried using $t^*=at$ but I seem to have a $b^3$ term on the right hand side then that I can't get rid of.

Then for the delay differential equation, it's similar but with $dN/dt = N(t)[a - bN(t-T)]$. He uses the above scalings, additionally replacing $T$ with $aT$. He got $dN(t)/dt = N(t)[1 - N(t-T)]$. I can't arrive at this solution, could somebody give me a step by step answer because all the things I've found online didn't help much. Thank you!

Ps. Really sorry for not having it in a better format, I couldn't figure out the code to make the fraction and powers :|

Multiply each side of the equation by the factor $b/a^2$
$$\mathcal{N} \stackrel{\rm def}{=} \frac{bN}{a} ~~~~\mbox{and}~~~~ \tau \stackrel{\rm def}{=} at$$