So Newton's law of cooling states that
$$ \frac{dT}{dt} = k(T - T_s) $$
And in my textbook it says that if we let $$y(t) = (T - T_s)$$
then $$ \frac{dy}{dt} = \frac{dT}{dt}$$
Which makes sense so far since $T_s$ is a constant
But then the book says
Thus our differential equation becomes...
$$ \frac{dy}{dt} = ky$$
Where in the world did $ky$ come from???
Newton's law of cooling would be $$\frac{dT}{d\color{red}{t}}=k(T-T_s)$$ so if we let $y(t)=T-T_s$ then we replace $T-T_s$ on the right-hand side of the equation with $y$ to get $$\frac{dT}{dt}=ky$$ and finally, we use replace $\frac{dT}{dt}$ with $\frac{dy}{dt}$ on the left-hand side of the equation (since these are equal as you say) to obtain $$\frac{dy}{dt}=ky.$$