Power series expansions usually work quite nicely within $t\in[-1,1]$ due to multiplication remaining in this interval.
Say we want to move some interval of interest $[a,b] \to [-1,1]$.
How can we algebraically manipulate a differential equation to squeeze a function into this area?
Some simple ones $f''(x) + f(x) = 0$ has famous solutions sin and cos. But these have long period, we would like to squeeze $[-\pi N,\pi N] \to [-1,1]$ so that we will be able to numerically calculate at least a few periods instead of just a fraction of a period.
The family of differential equations I want to consider are the same as in this previous question
$$\sum_{k=0}^{N} h_k(t)\cdot f^{(k)}(t) + g(t) = 0$$ Where $h_k(t)$s have some truncated power series expansion. $$h_k(t) = \sum_{k=0}^{M} c_kt^k$$ We can easily verify we will quickly get numerical issues when $M$ large and $|t|\gg1$.
So now to the question, given a known function expressed this way, how can we alter the $h_k(t)$s in order for the new solution to be linearly scaled $[a,b] \to [-1,1]$?