Oh, one more thing: you say, and I understand why, that this is "more or less pointless", but it's not! It is a perfectly good way of producing new functions. For some reason (alphabetical filing in my mind?), I always think first of the Airy function (https://en.wikipedia.org/wiki/Airy_function), but the Bessel functions (https://en.wikipedia.org/wiki/Bessel_function) and, more generally, most (all?) special functions (https://en.wikipedia.org/wiki/Special_functions) also arise in this way.
In fact, even the logarithmic function (boringly, via $\dot y = 1/x$) and the exponential and sine functions (more interestingly, via $\dot y = y$ and $\ddot y = -y$) can be defined this way (by imposing suitable initial conditions, once you know the relevant existence and uniqueness theorems). They can also be defined by their power series, without direct reference to differential equations, but I find such a definition hard to motivate without reference to the differential-equations definition.