What is the meaning of the word heuristically in this text?
the fuction
$$J_{n}(x)=x^{n}\sum\limits_{m=0}^{\infty}{{{(-1)^{m}x^{2m}}\over{2^{2m+n}m!(n+m)!}}}$$
is called the Bessel function of the first kind of order $n$. This series converges for all $x$, as the ratio test shows. Hence $J_{n}(x)$ is defined for all $x$.
Example: Bessel functions $J_{0}(x)$ and $J_{1}(x)$
For $n=0$ we obtain the Bessel function of order 0 $$J_{0}(x)=\sum\limits_{m=0}^{\infty}{{{(-1)^{m}x^{2m}}\over{2^{2m}(m!)^{2}}}}=1-{{x^{2}}\over{2^{2}(1!)^{2}}}+{{x^{4}}\over{2^{4}(2!)^{2}}}-{{x^{6}}\over{2^{6}(3!)^{2}}}+-\cdots$$
which looks similar to a cosine. For $n=1$ we obtain the Bessel function of order 1 $$J_{1}(x)=\sum\limits_{m=0}^{\infty}{{{(-1)^{m}x^{2m+1}}\over{2^{2m+1}m!(m+1)!}}}={{x}\over{2}}-{{x^{3}}\over{2^{3}1!2!}}+{{x^{5}}\over{2^{5}2!3!}}-{{x^{7}}\over{2^{7}3!4!}}+-\cdots$$
which looks similar to a sine. But the zeros of these functions are not completely regularly spaced and the height of the “waves” decreases with increasing $x$. Heuristically, ${{n^{2}}\over{x^{2}}}$ in $y''+{{1}\over{x}}y'+(1-{{n^{2}}\over{x^{2}}})y=0$ ($x^{2}y''+xy'+(x^{2}-n^{2})y=0$ divided by $x^{2}$) is small in absolute value for large $x$, and so is ${{1}\over{x}}y'$, so that then Bessel’s equation comes close to $y''+y=0$, the equation of $\sin x$ and $\cos x$; also ${{1}\over{x}}y'$ acts as a “damping term,” in part responsible for the decrease in height. One can show that for large $x$, $$J_{n}(x)\sim\sqrt{{{2}\over{\pi x}}}\cos(x-{{n\pi}\over{2}}-{{\pi}\over{4}})$$
where $\sim$ is read “asymptotically equal” and means that for fixed n the quotient of the two sides approaches $1$ as $x\rightarrow\infty$.
[''Advanced Engineering Mathematics'' by ''Erwin Kreyszig'' Page 189]
I consulted a number of available dictionaries and the possible meanings I found for the word heuristic are these: 1-(Mathematics) (of a method of solving problems) one for which no algorithm exists and which therefore depends on inductive reasoning from past experience of similar problems
2-proceeding to a solution by trial and error or by rules that are only loosely defined
Which of these meanings is correct (if any)? {I asked this in ELL and someone said it seems like more of a question for Math Stack Exchange than for ELL.}