Hello I am trying to show that the equation
$\frac{dN}{dt} = rN(1 - \frac{N(t - \tau)}{K})$
can be rewritten in a dimensionless form as
$\frac{dy}{dx} = \lambda y(1 - y(x - 1))$ using the transformations $y = \frac{N}{K}$ and $x = \frac{t}{\tau}$.
So far my attempt at the problem is to assume $\frac{dy}{dx} = \frac{dy}{dN}*\frac{dN}{dt}*\frac{dt}{dx}$. Using the given information of $y = \frac{N}{K}$ and $x = \frac{t}{\tau}$ it seems
$\frac{dy}{dN} = \frac{1}{K}$ and $\frac{dx}{dt} = \frac{1}{\tau} \Rightarrow \frac{dt}{dx} = \tau$
I then proceeded to substitute this information into $\frac{dy}{dx} = \frac{dy}{dN}*\frac{dN}{dt}*\frac{dt}{dx}$,
$\frac{dy}{dx} = \frac{1}{K}*(rN(1 - \frac{N(t - \tau)}{K}))*\tau = r\tau(\frac{N}{K}(1 - \frac{N}{K}(t - \tau)) =$
$r\tau y(1 - y(t - \tau))$
At this point I am unsure if I should try to think of a factor to multiply the last step of my simplification so that I can assume $\lambda$ is equal to some factor and then match it with what $\frac{dy}{dx}$ is supposed to be or if I made a mistake in how I approached the problem so far and it involves a more complicated understanding of how to apply the chain rule. Needless to say in my simplification it should be noted that the inside function of $y = \frac{N}{K}$ currently is in the same form of $(t - \tau)$ as it is in $\frac{dN}{dt}$ but by multiplying by $\frac{1}{\tau}$ the resultant inside will be of the form $(\frac{t}{\tau} - \frac{\tau}{\tau}) = (x - 1)$. So my question is what needs to be done to further proceed along this problem? Any help would be much appreciated, thanks.