4cosh3x-5sinh3x=3
You asked:
Solve the equation \(4 \cdot \cosh\left( 3 x \right) - 5 \cdot \sinh\left( 3 x \right) = 3\) for the variable \(x\).
MathBot Answer:
The real solution is: \[x = \frac{\log{\left(-3 + 3 \sqrt{2} \right)}}{3} \approx 0.072412901\]
The complex solutions are: \[\begin{aligned}x &= \left\{\frac{\log{\left(-3 + 3 \sqrt{2} \right)}}{3} + \frac{2 \pi i n}{3}\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{2.0943951 n i + 0.072412901\; \middle|\; n \in \mathbb{Z}\right\}\\x &= \left\{\frac{\log{\left(3 + 3 \sqrt{2} \right)}}{3} + \frac{i \left(\pi + 2 \pi n\right)}{3}\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{0.33333333 i \left(6.2831853 n + 3.1415927\right) + 0.65999529\; \middle|\; n \in \mathbb{Z}\right\}\end{aligned}\]
\(i\) is the imaginary unit, defined as \(i^2 = -1\).