I've been given a question that I can't find anything on or wrap my head around.
I think that in part a, because it separates to $f(y) = \exp(t+c)$ then $f(t-\tau) = \exp(t+c-\tau)$ and since the constant and tau are both arbitrary, they can be combined to ensure that it's still a solution. But I don't know how to use that in part c which makes me think I'm wrong.
Thank you for any help.
By the simple change of variable $x=\xi-\tau$, such that $dx=d\xi$,
$$\frac{dy(x)}{dx}=f(y(x))$$
becomes
$$\frac{dy(\xi-\tau)}{dx}=\frac{dy(\xi-\tau)}{d\xi}=f(y(\xi-\tau)),$$
which is formally equivalent to
$$\frac{dy(x-\tau)}{dx}=f(y(x-\tau)).$$