Today I was fiddling about with a TI-89 calculator, attempting as usual to confuse it. I figured that making it solve an equation with a periodic function would be fun, so I tried the following:
$\cos(\pi x) + x^2 = 0$
While this didn't stump it, I noted that the solution it gave was just a decimal. Since I ran this through solve(), which usually will give you a nice solution like $\frac{\sqrt{2}}{2}$ if it can, I found this rather interesting. I figured what it eventually did was give up on solving it in its usual manner and move to a numeric method: i.e., it simply made intelligent guesses until it found good solutions.
I am usually inclined to think most random decimals have a closed form expression behind them. (In fact, do all such decimals have a closed form, even if we don't know it? I may have to look into that.) As such, I decided to put this into Wolfram|Alpha and see if it had any better results. But no dice; it gave me back, similar to the TI-89, that $x = \pm 1$ and $x \approx \pm 0.629847$ were solutions.
The latter decimal, $x \approx 0.629847$, is the one I am concerned with. As far as I am concerned, $x = \pm 1$ are sort of "trivial" solutions; just thinking the problem over leads you to them naturally.
Is there a way to solve this algebraically? I can sort of narrow it down. I know that as $x \to \infty$, the $\cos(\pi x)$ term is essentially trivial compared to $x^2$. Given that $x^2>0$ for any $x \ne 0$, and given that $\cos(\pi x)$ has a range of $[-1, 1]$, it seems to me that whenever $x^2>1$, $\cos(\pi x)$ cannot pull down $x^2$ enough for it become zero.
So, it seems natural to me to think, then, that all solutions must lie where $x^2 \le 1$, viz., within the interval $[-1, 1]$. This narrows down the field significantly, but it still does not really help me with an algebraic solution. (However, I figure if I were to come across this in a real-world scenario, this would be a useful line of attack for a guess-and-test sort of deal.)
Another line of attack I attempted was to take the reverse approach: use the numerically attained solution to find a closed-form solution. I thought it may be an interesting number I simply had not learned about, so I tried to look up the decimal sequence in the OEIS, but to no luck: no such sequence was available.
I've sort of rambled, so here are my questions:
- Is there an algebraic solution to the above equation?
- Even if there is not, is there any way to figure out the closed form expression behind the decimal $x = 0.629847$? I don't even care if the expression has $\cos$ or $\sin$ in it.
I will be honest: I really don't even know where to start.