How to prove a transcendental equation is never zero or sometimes zero?

Question

I am an engineer by profession. Recently, while working on beams, I obtained the following transcendental equation: $$1-\cos(x)\cosh(x)+x^2\sin(x)\sinh(x)$$. I was wondering if this expression can ever become zero?

I could prove the case that it cannot be identically equal to zero (considering the first two terms together as one term and the $\sin(x)$term as the second term). But I could not prove it for general case. I tried using asymptotics to see what happens at very large and very small $x$ in hope that it may reveal something, but I am dead lost. Is there any way or approach to prove such things?

Thanking you in advance!

P.S.: I have graphed the function in MATLAB and MAPLE, but there seems to a numerical issue. The figure shows the graph crossing zero, but substituting that value of $x$ in the function produces non-zero value.

Answer 1

Since:

• $f$ is continuous;
• $f\left(\frac\pi2\right)=1+\frac14\pi^2\sinh\left(\frac\pi2\right)>0$;
• $f\left(\frac{3\pi}2\right)=1-\frac94\pi^2\sinh\left(\frac{3\pi}2\right)<0$,

$f$ has a zero in the interval $\left(\frac\pi2,\frac{3\pi}2\right)$, by the intermediate value theorem.

Answer 2

Because of the last term of the equation, the zero's of function $$f(x)=1-\cos(x)\cosh(x)+x^2\sin(x)\sinh(x)$$ will be "close" to $n\pi$.

If you want to approximate them, expand as a Taylor series $f(x)$ around $x=n \pi$ to some order and use series reversion to obtain for example $$x_{(n)}=n\pi+\frac {\cosh (\pi n)-(-1)^n } {n^2\pi ^2 -1} \text{csch}(\pi n)+$$ $$n\pi\frac {\left((-1)^n \cosh (\pi n)-1\right)^2 (\pi n \coth (\pi n)+2)} {(n^2\pi ^2 -1)^3}\text{csch}^2(\pi n)+\cdots$$ For the first roots $$\left( \begin{array}{ccc} n & \text{estimate} & \text{solution} \\ 1 & 3.236938496 & 3.244010785 \\ 2 & 6.308170276 & 6.308218339 \\ 3 & 9.436006897 & 9.436010404 \\ 4 & 12.57269612 & 12.57269669 \\ 5 & 15.71201387 & 15.71201400 \\ 6 & 18.85236951 & 18.85236955 \\ 7 & 21.99321595 & 21.99321596 \\ 8 & 25.13432416 & 25.13432417 \\ 9 & 28.27558465 & 28.27558465 \\ 10 & 31.41693968 & 31.41693968 \end{array} \right)$$