Solving $\cosh(zx) = yx$?

Question

I am currently facing a problem where I have to solve the equation where $x\neq 0$:

$$\cosh(3.8x)=3.5x$$

Wolframalpha only outputs an array of numbers without much too it.

Does anybody know how i can resolve it? I cant just simply use cosh-1 because of the fact that i have x on both sides.

Answer 1

HINT: Change in exponential form $$\cosh(3.8x)=\frac{e^{3.8x}+e^{-3.8x}}{2}$$

Answer 2

Letting $y = 3.8x$, this is $\cosh(y) = ay $ with $a = 3.5/3.8 $ or $\dfrac{\cosh(y)}{y} =a $.

Since $\dfrac{\cosh(y)}{y} \gt 1.5 $ for $x \ge 0$ (according to Wolfy), there are no positive real solutions.

Answer 3

You want to find the zero of function $$f(x)=\cosh \left(\frac{19 x}{5}\right)-\frac{7 x}{2}$$ for which $$f'(x)=\frac{19}{5} \sinh \left(\frac{19 x}{5}\right)-\frac{7}{2}\qquad \text{and} \qquad f''(x)=\frac{361}{25} \cosh \left(\frac{19 x}{5}\right)\quad > 0 \quad \forall x$$ The first derivative cancels at $$x_*=\frac{5}{19} \sinh ^{-1}\left(\frac{35}{38}\right)\implies f(x_*)=\frac{1}{38} \left(\sqrt{2669}-35 \sinh ^{-1}\left(\frac{35}{38}\right)\right)\approx 0.600$$

The second derivative test shows that $x_*$ corresponds to a minimum and since $f(x_*)>0$, there is no real root.

If we look for the complex solution, first let $x=\frac{ 5}{19}y$ and then $y=a+i b$. Expanding, we need to solve $$\cosh (a) \cos (b)+\frac{35 a}{38}+i \left(\sinh (a) \sin (b)+\frac{35 b}{38}\right)=0$$ From the real part, we have $$b=\cos ^{-1}\left(-\frac{35a}{38} \text{sech}(a)\right)$$ Plug in the expression of the imaginary part.

So, we now need to solve for $a$ $$\sinh (a) \sqrt{1444-1225 a^2 \text{sech}^2(a)}+35 \cos ^{-1}\left(-\frac{35a}{38} \text{sech}(a)\right)=0$$ A quick plot reveals that the solution is close to $-1$; using this approximation, this gives $b_\pm\approx 0.93$.

Now, use Newton method with $x_0=-\frac 5{19}(1-i)$. The iterates will be $$\left( \begin{array}{cc} n & x_n \\ 0 & -0.263158+0.263158\, i \\ 1 & +0.070690+0.218730\, i \\ 2 & +0.248807+0.202118\, i \\ 3 & +0.242215+0.252746\, i \\ 4 & +0.245420+0.248767\, i \\ 5 & +0.245466+0.248783\, i \end{array} \right)$$