Jtdylktuy

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 $frac{dy}{dx} = frac{dy}{dN}*frac{dN}{dt}*frac{dt}{dx}$,

$frac{dy}{dx} = frac{1}{K}*(rN(1 - frac{N(t - tau)}{K}))*tau = rtau(frac{N}{K}(1 - frac{N}{K}(t - tau)) =$

$rtau y(1 - y(t - tau))$

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.